REVIEWS  OF 

RATIONAL  GEOMETRY 

BY 

GEORGE  BRUCE  HALSTED 

A  B.  and  A.  M.  (Princeton  University);  Pli.  D.  (Johns 
Hopl<ins  University);  F.  R.  A.  S.;  Ex-Fellow  of  Prince- 
ton University;  twice  Fellow  of  Johns  Hopkins  Univer- 
sity;  Intercollegiate  Prizeman  ;  sometime  Instructor  in 
Post-Graduate  Mathematics.  Princeton  University ;  Mem- 
ber of  the  American  Mathematical  Society  ;  Member  of 
the  London  Mathematical  Society;  Member  of  the  Society 
for  the  Promotion  of  Engineering  Education  ;  Member  of 
the  Mathematical  Association  ;  President  of  the  princj^tpn 
Alumni  Association  of  Texas;  Fellow  and  Past-P'resident 
of  the  Texas  Academy  of  Science  ;  Professor  of  Mathe- 
matics in  Kenyon  College;  Vice-President  of  i\iz  Ameri- 
can Association  for  the  Advancement  of  Scfjence,  and 
Chairman  of  Section  A  (Mathematics  and  Astronomy))' 
Non-Resident  Member  of  the  Washington  Academy  of 
Sciences;  Member  of  the  Society  of  Arts;  Membte  d'Hori- 
neur  du  Comite  Lobachefsky ;  Miembro  de  la  Sociedad 
Cientifica  "Alzate"  de  Mexico  ;  Socio  Corresponsal  de  la 
Sociedad  de  Geografia  y  Estadistica  de  Mexico;  Alitglied 
des  Vereins  zur  Foerderung  des  Unterrichts  in  der  Mathe- 
matik  und  den  Naturwissenschaften ;  Mitglied  der  D.eut- 
schen  Mathematiker-Vereinigung ;  Societaire  Perpetual  de 
la  Societe  Mathematique  de  France ;  Socio  Perpetuo  del 
Circolo  Matematico  di  Palermo. 


920387 


RATIONAL  GEOMETRY 


Rational  Geometry,  a  Text-book  for  the  Science  of 
Space,  par  George  Bruce  HalsTED.  —  Un  vol. 
in  12,  VIII  -I-  285  pages,  247  figures.  John 
Wiley  &  Sons,  Newyork,  1904. 

Les  recents  et  si  remarquables  travaux  de  M. 
Hilbert  sur  les  fondements  de  la  geometrie,  ma- 
gistralement  analyses  par  M.  Poincare  dans  ses 
articles  de  la  Revue  des  Sciences  et  dans  son  Rap- 
port sur  le  y  concours  du  prix  Lobatschefsky 
[1903],  ne  pouvaient  manquer  a  bref  delai 
d'eveiller  I'attention  des  geometres  et  d'exercer 
une  influence  profonde  et  decisive  sur  leurs  ouv- 
rages.  On  devait  certainement  s'attendre  a  voir 
publier  des  Traites  didactiques  dont  les  hardis  et 
erudits  auteurs,  rompant  resolument  avec  les  habi- 
tudes et  traditions  de  plus  de  vingt  siecles,  essaier- 


4  HALSTED'S  RATIONAL  GEOMETRY 

aient  d'harmc3niser*l"enSe'i^ement  de  la  geometrie 

avec  It/s'Jdpc^s jVji^yelifs.,.  jMa'lgje.-  que    M.  Hilbert 

eut    pris    deja    lui-meme    soin    d'indiquer    et    de 

jalonner  d'une  maniere  precise  la  route  a    suivre, 

la  taclie  etait  loin  d'etre  aisee.     Elle  devait  attirer 

particulierement  M.  George  Bruce  Halsted,  le 
savant  professeur  de  Kenyon  College,  un  des  plus 

ardents  defenseurs  de  la  geometrie  generale  aux 
Etats  Unis,  bien  connu  par  ses  nombreuses  publi- 
cations dans  les  Revues  "Science"  et  "z/lmcricau 
Matlieniaticjl  Moutlily" ,  et  surtout  par  ses  belles 
traductions  anglaises  de  Saccheri,  Bolyai  et  Lobat- 
schetsky.  La  "Rational  geometry"  de  M.  Hal- 
sted, encouragee  par  M.  Hilbert,  marque  une 
epoque  dans  I'histoire  des  livres  destines  a  I'en- 
seignement.  Nous  alions  analyser  en  detail  les 
chapitres  de  cet  ouvrage. 

Pour  constituer  une  geometrie  vraiment  ration- 
nelle,  deux  choses  etaient  indispensables  :  en 
premier  lieu,  t^tablir  une  liste  complete  des  axiomes 
en  s'efforcant  de  n't^w  oublier  aucun;  ensuite, 
supprimer  totalement   le   rule   de   I'intuition  qui  a 


HALSTED'S  RATIONAL  GEOMETRY  '-        5 

occupe  jusqu'ici  Line  place  telle  en  geometrie  que 
nous  faisons  dans  cette  science  presque  a  chaque 
instant  usage  de  propositions  intuitives  sans  nous 
en  apei'cevoir  le  moins  du  monde.  Dans  ce  but, 
les  axiomes  qui  expriment  les  relations  mutuelles 
pouvant  exister  entre  les  etres  geometriques,  point, 
droite,  plan,  espace,  ont  ete  suivant  la  methode 
de  M.  Hilbert,  repartis  en  cinq  groupes:  Con- 
nexion ou  association,  ordre,  congruence,  axiome 
des  paralleles  ou  d'Euclide,  axiome  d'Archimede 
ou  de  continuite. 

Dans  le  chapitre  I,  M.  Halsted  definit  les  etres 
geometriques  et  expose  les  sept  axiomes  de  con- 
nexion. De  ces  axiomes  decoulent  naturellement 
les  propositions  habituelles. 

Deux  droites  distinctes  ne  peuvent  avoir  deux 
points  cummuns. 

Deux  droites  distinctes  ont  un  point  commun 
ou  n'en  ont  aucun. 

Deux  plans  distincts  ont  en  commun  une  droite 
ou  n'ont  aucun  point  commun. 

Un  plan  et  une  droite  qui  n'y  est  pas  situee  ont 
un  point  commun  ou  aucun. 


6  HAJ.STED'S  RATIONAL  GEOMETRY 

Par  une  druite  et  un  point,  ou  dt-iix  droites  qui 
ont  un  puint  commun,  on  peut  faire  passer  un 
plan  et  un  seul. 

—  DanP  le  chapitiv  11    viennent,  au  nombre  de 
quartrp,  ItfS  axiomes  de  I'ordre   qui  precisent  I'ar- 
rangeitiiBnt  des  points  caracterise  par  le  mot  enti'e. 
Ces  axlomes  sont  completes    par   la    definition  du 
segnient    qui    ne    doit   eveiller   aucune    idee    de 
mesure:   Deux  points  A  et    B  de  la  droite  A  defin- 
issept    le    sejiment    AB   ou    BA;  les   points  de  la 
driiite  sitiies  entrc   A   et  B  sont   les  points  du  seg- 
ment.      De     la    la    distinction    entre    les     deux 
I'ciyotis      d'une      droite    separes    par     un     point, 
entre    les     deux    regions     du    plan    separees    par 
une  droite.  —  Points  interieurs  et  exterieurs  a  un 
polygone.  —  Notons   pour  memoire  I'axiome  4  ou 
axiome  de  Pasch.      Si  A,  B  it  C  sont  trois  points 
non  colIint'Liircs  ct  a   ////<'  droite  du  plan  nc  passant 
par  aiiiiin  d'eux,  lorsque  a  renfermc  un  point   du 
segment  A  B,  elle  en  a  un  autre  sur  B  C  ou  sur  A  C. 
11  est  evident  que  si  le  plus  petit  role  etait  laisse  a 
I'intuition,    on    ne    songerait  pas    a    enoncer  cette 


HALSTED'S  RATIONAL  GEOMETRY  7 

proposition  dont  on  fait  inconsciemment  un  si  fre- 
quent usage. 

Le  chapitre  111  develuppe  les  axiomes  de  con- 
gruence :  segments,  angles,  triangles,  et  I'auteur 
y  formule  en  ces  termes  precis  le  theoreme  general 
de  congruence. 

5/  A  B  C...  A'  B'  C...  sont  deux  figures  con- 
gruentes,  et  que  P  designe  un  point  quelconque 
de  la  premiere,  on  peut  toujours  trouver  de  fagon 
univoque  dans  la  deuxieme  un  Point  P'  tel  que 
les  figures  ABC...  P,  A'B'C"  ...  P'  soient  con- 
gruentes. 

Ce  theoreme  exprime  I'existence  d'une  certaine 
transformation  unique  et  reversible  qui  nous  est 
familiere  sous  le  nom  de  deplacement.  La  notion 
de  deplacement  est  done  bassee  sur  celle  de  con- 
gruence, ce  qui  est  absolument  logique. 

Le  chapitre  suivant  est  consacre  a  I'axiome  de 
la  parallele  unique  et  aux  propositions  qui  en  sont 
la  consequence.  La  plupart  sont  classiques,  nous 
n'y  insistons  pas;  mais  il  en  est  d'autres  que  nous 
avons  eu  jusqu'ici  I'habitude  de  considerer  comme 


HALSTED'S  RATIONAL  GEOMETRY 


intuitives  et  qui  ne  le  sont  pas.  M.  Halsted  les 
demontie  avec  raison;  ce  sont  celles-ci:  Tout 
segment  a  un  point  milieu;  tout  angle  a  un  rayon 
bissecteur. 

Chapitre  V  — Circonterence. 

Chapitre  VI  —  Problemes  de  Construction. 
Toutes  les  constructions  decoulant  des  theoremes 
bases  sur  les  cinq  groupes  d'axiomes  peuvent  etre 
graphiquement  resolues  par  la  regie  et  le  trans- 
porteur  de  segments  (Streckeniibertrager  de  M. 
Hilbert)  et  ramenees  a  ces  deux  traces  fonda- 
mentaux:  Tracer  une  droite;  prendre  sur  une 
droite  donnee  un  segment  donne. 

Chapitre-  VII  —  Egalites  et  inegalites  entre 
cotes,  angles  etarcs. 

Chapitre  VIII  —  Calcul  des  Segments.  En  se 
basant  sur  les  axiomes  des  groupes  I,  II,  IV  et  en 
mettant  systematiquement  de  cote  Taxiome 
d'Archimede  dont  on  s'est  passe  dans  ce  qui 
precede  et  dont  on  peut  egalement  se  passer  dans 
ce  qui  suit,  on  arrive  a  creer,  independamment  de 
toute  preoccupation    metrique,    un    calcul    de  seg- 


HALSTED'S  RATIONAL  GEOMETRY  ^> 

merits  ou  les  operations  sont  identiques  a  celles 
des  nombres.  Sommes  et  produits  de  segments. 
Sommes  d'arcset  d'angles. 

Chapitre  IX.  —  Proportions  et  similitudes. 
Deux  triangles  sont  dits  semblables  quand  leurs 
angles  sont  respectivement  congruents.  11  eutfallu 
dire  la  un  mot  de  IVxistence  de  tels  triangles;, 
c'est  une  lacune  bien  facile  a  combler.  La  simili- 
tude conduit  naturellement  au  tbeoreme  de  Thales 
et  aux  proportionnalites  qui  t-n  decoulent. 

Chapitre  X  —  Equivalence  dans  le  plan.  La 
mesure  des  aires  planes  peut  etre  obtenue  sans 
le  secours  de  I'axiome  d'Archimede  parce  que 
deux  polygones  equivalents  peuvent  etre  con- 
sideres  comme  sommes  algebriques  de  triangles 
elementaires  en  meme  nombre  et  deux  a  deux  con- 
gruents, quoique  de  dispositions  differentes.  Par 
definition  I'aire  d'une  triangle  egale  le  demi 
produit  de  la  base  par  la  hauteur;  deux  polygones 
equivalents  ont  meme  aire  et  reciproquement. 
Theoreme  de  Pythagore  et  carres  construits  sur 
les  cotes   d'un   triangle.     Le  chapitre  se  termine 


10  H^LSTED'S   RATIONAL  GEOMETRY 

par   Line  note  historique  courte,  mais  interessante 
SLir  le  numbre  tt. 

Chapitre  XI  —  Geometrie  du  plan,  differant 
pen  de  notre  cinquieme  livre  usuel. 

Le  chapitre  XI 1  est  consacre  aux  polyedres  et 
volumes.  M.  Halsted  commence  a  bon  droit  par 
le  theoreme  d'Euler;  il  appelle  par  dcuififion 
Volume  du  tetraedre  le  tiers  du  produit  de  la  base 
par  la  hauteur,  et  prouve  que  le  volume  d'un 
tetraedre  egale  la  somme  des  volumes  des  tetrae- 
dres  en  lesquels  on  le  partage  d'une  fagon  quel- 
conque.  L'auteur  examine  quatre  methodes  de 
division  particulieres,  la  division  la  plus  generale 
pent  etre  obtenue  au  moyen  de  ces  dernieres,  et 
il  en  est  de  meme  pour  un  polyedre. 

Les  chapitres  XIII  et  XIV  nous  donnent  I'etude 
de  la  sphere,  du  cylindre  et  du  cone,  avec  le 
mesure  de  leurs  surfaces  et  volumes.  Pour  le 
volume  de  la  sphere.  Ton  fait  usage  de  I'axiome 
de  Cavalieri:  Si  deux  solides  compris  entre 
deux    plans    paralleles    sont   coupes   par   un    plan 


HALSTED'S  RATIONAL  GEOMETRY  11 

quelconque  parallele  aux  deux  premiers    suivants 
des  aires  e;c^ales,  ils  ont  meme  volume. 

Chapitre  XV  Spherique  pure  ou  Geometrie  a 
deux  dimensions  sur  la  sphere:  Ce  Chapitre  ne 
pouvait  manquer  de  trouver  ici  sa  place.  M. 
Halsted  y  precise  d'abord  ce  que  devinnent  a  la 
surface  de  la  sphere  les  axiomes  d 'association, 
d'ordre  et  de  congruence,  il  en  deduit  simplement 
et  naturellement  les  proprietes  elementaires,  trop 
negligees  dans  I'ensignement,  des  triangles 
spheriques. 

Trois  notes  terminent  I'ouvrage,  et  sont  rela- 
tives; Tune  a  theoreme  de  Tordre,  la  deuxieme 
au  compas,  et  la  troisieme  ci  la  solution  des  pro- 
blemes. 

Ainsi  qu'on  le  voit  par  cette  analyse,  le  livre  de 
M.  Halsted  constitue  une  innovation  et  une  tenta- 
tive de  vulgarisation  des  plus  interessantes.  Pour 
lui  donner  plus  de  poids  aupres  des  etudiants  a 
qui  il  est  destine,  I'eminent  professeur  de  Kenyon 
College  y  a  ajoute  700  exercices  formant  un  choix 


12  HALSTED'S  RATIONAL  GEOMETRY 


excellent  et  varie.     Nous  souhaitons  a  cet  ouvrage 

de  notre  distingue  ami   tout  le  succes  qu'il  merite. 

P.  Barbarin. 

President  de  la  Societe  des  Sciences  physiques 
et  naturelles  de  Bordeaux. 

I'Enseignement  JWathematique 
du  15  Mars  1905. 


HALSTED'S  RATIONAL  GEOMETRY  13 


SCIENTIFIC  BOOKS. 

Rational  Gt'omctiy.  By  GEORGE  BRUCE  Hal- 
STED.  New  York  and  London,  John  Wiley 
and  Sons.     1904.     Pp.  viirr285. 

For  over  two  thousand  years  there  has  been 
only  one    authoritative    text-hook    in    geometry. 

"  No  text-book,"  says  tlie  British  Association, 
"that  has  yet  been  produced  is  fit  to  succeed 
Euclid  in  the  position  of  authority!"  There  is,  in 
fact,  little  improvement  to  be  made  in  Euclid's 
work  along  the  lints  w'.iich  he  adopted,  and 
among  the  multitude  of  modern  text-books,  each 
has  fallen  under  the  weight  of  criticism  in  pro- 
portion to  its  essential  deviation  from  that  ancient 
autlvjr. 

This  does  not  mean  tiiat  Euclid  is  witliout 
defect,  but  starting  from  his  discussion  of  his 
famous  parallel  postulate,  the  modern  develop- 
ment has  been  in  the  direction  of  the  extension 
of  geometrical  science,  with  the  place  of  that 
author  so  definitely  fixed  that  the  system  which 
lie  developed  is  called  Euclidean  geometry,  to 
distinguish     it  from  new  developments.     The  de- 


14         HALSTED'S  RATIONAL  GEOMETRY 

fects  of  Euclid  arise  out  of  a  new  view  of  rigorous 
logic  whose  objections  seem  finely  spun  to  the 
average  practical  man,  but  which  are  based  upon 
sound  thought.  The  key  to  this  modern  criticism 
is  the  doubt  which  the  mind  casts  upon  the  relia- 
bility of  the  intuitions  of  our  senses,  and  the 
tendency  to  make  pure  reason  the  court  of  last 
resort.  Thus,  the  sense  of  point  between  points, 
the  perception  of  greater  and  less  and  many  other 
tacit  assumptions  of  the  geometrical  diagram,  are 
the  vitiating  elements  on  which  modern  criticism 
concentrates  its  objections. 

As  an  evidence  of  the  ease  with  which  the 
senses  can  be  made  to  deceive,  take  a  triangle 
ABC,  in  which  AC  is  slightly  greater  than  BC. 
Erect  a  perpendicular  to  AB  at  its  middle  point  to 
meet  the  bisector  of  the  angle  C  in  the  point  D, 
From  D  draw  perpendiculars  to  AC,  BC,  meeting 
them  respectively  in  the  points  E,  F.  Let  the 
senses  admit,  as  they  readily  will  in  a  free-hand 
diagram,  that  E  is  between  A  and  C,  and  F 
between  B  and  C;  then  fmm  the  equal  right 
triangles  AED=BFD,  DEC  =  DEC,  we  find 
AE=BF,  EC=FC,  and,  by  adding,  AC  =  BC, 
whereas  AC  is  in  fact  greater  than  BC. 


HALSTED'S  RATIONAL  GEOMETRY  15 


Are  we  to  take  our  eyes  as  evidence  that  one 
point  lies  between  two  other  points,  or  how  are 
we  to  establish  that  tact?  This  query  alone  lets 
in  a  flood  of  criticism  on  all  established  demon- 
strations. The  aim  of  modern  rational  geometry 
is  to  pass  from  premise  to  conclusion  solely  by 
the  force  of  reason.  Points,  lines  and  planes  are 
the  names  of  things  which  need  not  be  physically 
conceived.  The  object  is  to  deduce  the  conclu- 
sions which  follow  from  certain  assumed  rflations 
between  these  things,  so  that  if  the  relations  hold 
the  conclusions  follow,  whatever  these  things 
may  be.  Space  is  the  totality  of  these  things; 
its  properties  are  solely  logical,  and  varied  in 
character  according  to  the  assumed  fundamental 
relations.  Those  assumed  relations  which  de- 
velop space  concepts  that  are  apparently  in  accord 
with  vision  constitute  the  modern  foundations  of 
Euclidean  space. 

Mr.  Halsted  is  the  first  to  write  an  elementary 
text-book  which  adopts  the  modern  view,  and  in 
this  respect,  his  "  Rational  Geometry  "  is  epoch- 
making.  It  is  based  upon  foundations  which 
have  been  proposed  by  the  German  mathema- 
tician, Hilbert.      in    point   of  fact,   the   book  con- 


16         HALSTED'S  RATIONAL  GEOMETRY 

tains  numerous  diagrams,  and  is  not  to  be  dis- 
tinguished in  this  respect  from  ordinary  text- 
books, but  these  are  simply  gratuitous  and  not 
necessary  accompaniaments  of  tlie  argument,  de- 
signed especially  for  elementary  students  whose 
minds  would  be  unequal  to  the  task  of  reveling  in 
the  domain  of  pure  reason.  Also,  in  opening  the 
book  at  random,  one  does  not  recognize  any  great 
difference  from  an  ordinary  geometry.  In  other 
words,  those  assumed  relations  are  adopted  which 
lead  to  Euclidean  geometry,  in  this  respect  the 
author  is  appealing  to  the  attention  of  elementary 
schools,  where  no  geometry  other  than  the  prac- 
tical geometry  of  our  world  has  a  right  to  be 
taught. 

The  first  chapter  deals  with  the  first  group  of 
assumptions,  the  assumptions  of  association. 
Thus,  the  first  assumption  is  that  hco  iiistinct 
poUits  determtne  a  stniio/it  line.  This  associates 
two  things  called  points  with  a  thing  called  a 
straight  line,  and  is  not  a  definition  of  the  straight 
line.  The  definition  of  a  straight  line  as  the 
shortest  distance  between  two  points  involves  at 
once  an  unnamed  assumption,  the  conception  of 
distance,    which     is    a    product   of  our    physical 


HALSTED'S  RATIONAL  GEOMETRY  17 

senses,  whereas  the  rational  development  of  ge- 
ometry seeks  the  assumptions  which  underlie  and 
are  the  foundations  of  our  physical  senses.  hi 
the  higher  court  of  pure  reason,  the  testimony  of 
our  physical  senses  has  heen  ruled  out,  not  as 
utterly  incompftent,  hut  as  not  conforming  to  the 
legal  requirements  of  the  court.  However,  there 
is  no  ohjection  to  shortness  in  names,  and  a 
straight  line  is  contracted  into  a  straioht,  a  seg- 
ment of  a  straight  line,  to  a  sect,  etc. 

In  the  second  chapter  we  find  the  second  group 
of  assumptions,  the  assumptions  of  betweenness, 
which  develop  this  idea  and  the  related  idea  of 
the  arrangement  of  points,  hi  the  next  chapter 
we  have  a  third  group,  the  assumption  of  congru- 
ence. This  chapter  covers  very  nearly  the 
ordinary  ground,  with  respect  to  the  congruence 
of  angles  and  triangles,  and  all  the  theory  of 
perpendiculars  and  parallels  which  does  not 
depend  upon  Euclid's  famous  postulate.  This 
postulate  and  its  consequences  are  considered  in 
chapter  IV. 

All  the  school  propositions  of  both  plane  and 
solid  geometry  are  eventually  developed,  although 
there  is  some  displacement  in  the  order  of    propo- 


18  HALSTED'S  RATIONAL  GEOMETRY 

sitions,  due  to  the  method  of  development. 
Numerous  exercises  are  appended  at  tlie  end  of 
chapters,  which  are  numbered  consecutively  from 
1  to  700. 

Undoubtedly  the  enforcement  upon  logic  of  a 
a  blindness  to  all  sense  perceptions  introduces 
some  difficulties  which  the  ordinaiy  cjeometries 
seem  to  avoid,  but  as  in  the  case  of  our  concep- 
tfon  of  a  blind  justice,  this  has  its  compensation 
in  the  greater  weight  of  her  decisions.  It  seems 
as  if  the  present  text-book  (uight  not  to  be  above 
the  heads  of  the  average  elementary  students,  and 
that  it  should  serve  to  develop  the  logical  power 
as  well  as  practical  geometrical  ideas.  Doubtless, 
some  progressive  teachers  will  be  found  who  will 
venture  to  give  it  a  trial,  and  thus  put  it  to  the 
tests  of  experience.  At  least  the  work  will  appear 
as  a  wholesome  contrast  to  many  elementary 
geometries  which  have  been  constructed  on  any 
fanciful  plan  of  plausible  logic,  mainly  with  an 
eye  to  the  chance  of  profit. 

Arthur  S.  Hathaway, 
rose  polytechnic  instisute. 

ISCIHNCE,  Feb.  3,  1905.'] 


HALSTED'S  RATIONAL  GEOMETRY  19 


HALSTED'S    RATIONAL    GEOMETRY. 

''liatioiia/  Geo})h'try,  a  Text-book  for  the  Science 
of  Space.  By  GEORGE  BRUCE  HALSTED. 
New  York,  John  Wiley  &  Sons  (London, 
Chapman  &  Hall,  Limited).     1904. 

In  his  review  of  Hilbert's  Foundations  of  Ge- 
ometry, Professor  Sommer  expressed  the  hope 
that  the  important  new  views,  as  set  forth  by 
Hilbert,  might  be  introduced  into  the  teaching  of 
elementary  geometry.  This  the  author  has  en- 
deavored to  make  possible  in  the  book  before  us. 
What  degree  of  success  has  been  attained  in  this 
endeavor  can  hardly  be  determined  in  a  brief  re- 
view but  must  await  the  judgment  of  experience. 
Certain  it  is  that  the  more  elementary  and  funda- 
mental parts  of  the  "  Foundations  "  are  here  pre- 
sented, for  the  first  time  in  English,  in  a  form 
available  for  teaching. 

The  author's  predisposition  to  use  new  terms, 
as  exhibited  in  his  former  writings,  has  been  ex- 
hibited here  in  a  marked  degree.  Use  is  made  of 
the  terms  sect  for  segment,  straight  in  the  mean- 


20  HALSTED'S  RATIONAL  GEOMETRY 

ing  of  straiglit  line,  betweenness  instead  of  order, 
copunjtal  for  concurrfnt,  costraight  for  collinear, 
inversely  for  conversely,  assumption  for  axiom, 
and  sect  calculus  instead  of  algebra  of  segments. 
Not  the  slightest  ambiguity  results  from  any  of 
these  substitutions  for  the  more  common  terms. 
The  use  of  sect  for  segment  has  some  justifica- 
tion in  the  fact  that  segment  is  used  in  a  different 
sense  when  taken  in  connection  with  a  circle. 
Sect  could  well  be  taken  for  a  piece  of  a  straight 
line  and  segment  reserved  for  the  meaning  usu- 
ally assigned  when  taken  in  connection  with  a 
circle. 

The  designation,  betweenness  assumptions, 
which  expresses  more  concisely  the  ci^ntent  of 
the  assumptions  known  as  axioms  of  order  in  the 
translation  of  the  "Foundations"  of  Hilbert,  is 
decidedly  commendable.  As  motion  is  to  be  left 
out  of  the  treatment  altogether,  copunctal  is  bet- 
ter than  concurrent.  Permitting  the  substitution 
of  straight  for  straight  line,  then  costraight  is  pre- 
ferable to  collinear.  hiversely  should  not  be  sub- 
stituted for  conversely.  The  meaning  of  the 
latter  given  in  the  Standard  Dictionary  being 
accepted  in  all  mathematical  works,  it  is  well  that 


HALSTED'S  RATIONAL  GEOMETRY  21 

it  should  stand.  The  term  axiom*  lias  been  used 
in  so  many  different  ways  in  matliematics  tliat  it 
seems  best  to  abandon  its  use  altogether  in  pure 
mathematics.  The  substitution  of  assumption  for 
axiom  is  very  acceptable  indeed. 

The  first  four  chapters  are  devoted  to  statements 
of  the  assumptions  and  proofs  of  a  few  important 
theorems  which  are  directly  deduced  from  them. 
The  proof  of  one  of  the  betweenness  theorems 
(§29),  that  every  simple  polygon  divides  the  plane 
into  two  parts  is  incomplete,  as  has  been  pointed 
out,t  yet  the  proof  so  far  as  it  goes,  viz.,  for  the 
triangle,  is  perfectly  sound.  It  is  so  suggestive 
that  it  could  well  be  left  as  an  exercise  to  the  stu- 
dent to  carry  out  in  detail.  The  fact  that  Hilbert 
did  not  enter  upon  the  discussion  of  this  theorem 
is  no  reason  why  our  author  should  not  have  done 
so.     Hubert's  assumption  V,  known  as  the  Archi- 

*  "  The  familiar  definition:  An  axiom  is  a  self-evident 
truth,  means  if  it  means  anythins:,  that  the  proposition 
which  we  call  an  axiom  has  been  approved  by  us  in  the 
light  of  our  experience  and  intuition.  In  this  sense  ma- 
thematics has  no  axioms,  for  mathematics  is  a  tormal 
subject  over  which  formal  and  not  material  implication 
reigns."  E.  B.  Wilson,  BULLETIN,  Vol.  ii,  Nov.,  1904, 
p.  81. 

tDehn,  Jahresbericht  d.  Deutschen  Math.-Vereinigung, 
November,  1904,  p.  592. 


22  HALSTED'S  RATIONAL  GEOMETRY 


medes  assumption,  part  of  the  assumption  of 
continuity  whicli  our  author  carefully  avoids  using 
in  the  development  of  his  subject,  is  placed  at  the 
end  of  Chapter  V,  in  which  the  more  useful  pro- 
perties of  the  circle  are  discussed.  For  the  be- 
ginner in  the  study  of  demonstrative  geometry,  it 
has  no  place  in  the  text.  For  teachers  and  former 
students  of  Euclid  who  will  have  to  overcome 
many  prejudices  in  their  attempts  to  comprehend 
the  nature  of  the  "important  new  views"  set 
forth  in  the  "  Foundations"  it  has  great  value  by 
way  of  contrast.  Contrary  to  Sommer's  state- 
ment in  his  review  of  the  "  Foundations  "  (see 
Bulletin,  volume  6,  page  290)  the  circle  is  not 
defined  by  Hilbert  in  the  usual  way.  It  is  defined 
by  Hilbert  and  likewise  by  Halsted  according  to 
the  common  usage  of  the  term  circle.  The  defi- 
nition is — if  C  be  any  point  in  a  plane  a,  then  the 
aggregate  of  all  points  A  in  a,  for  which  the  sects 
CA  are  congruent  to  one  another,  is  called  a  circle. 
The  word  circumference  is  omitted  entirely,  with- 
out loss. 

In  the  chapter  on  constructions  we  have  a  dis- 
cussion of  the  double  import  of  problems  of  con- 
struction.    The  existence  theorems  as   based   on 


HALSTED'S  RATIONAL  GEOMETRY  23 

assumptions  I — V  are  shown  to  be  capable  of  gra- 
phic representation  by  aid  of  a  ruler  and  sect-car- 
rier. In  this  the  reader  may  mistakenly  suppose 
on  fust  reading  that  the  author  had  made  use  of 
assumption  V,  but  this  is  not  the  case.  While  in 
the  graphic  representation  the  terminology  of  mo- 
tion is  freely  used,  it  is  to  be  noted  that  the  ex- 
istence theorems  themselves  are  independent  of 
motion  and  in  fact  underlie  and  explain  motion. 
The  remarks,  in  §157,  on  the  use  of  a  figure,  form 
an  excellent  guide  to  the  student  in  the  use  of  this 
important  factor  in  mathematical  study.  In  chap- 
ter VIII  we  find  a  discussion  of  the  algebra  of 
segments  or  a  sect-calculus.  The  associative  and 
commutative  principles  for  the  addition  of  seg- 
ments are  established  by  means  of  assumptions 
IIlj  and  III^.  To  define  geometrically  the  pn)duct 
of  two  sects  a  construction  is  employed.  At  the 
intersection  of  two  perpendicular  lines  a  fixed 
sect,  designated  by  1,  is  laid  off  on  one  from  the 
intersection,  a  and  b  are  laid  off  in  opposite  senses 
on  the  other.  The  circle  on  the  free  end  points 
of  1,  a  and  b  determines  on  the  fourth  ray  a  sect 
c  =  ab.  This  definition  is  not  so  good  as  the  one 
given  by  the  "Foundations,"  as  it  savors  of  the 


24  HALSTED'S  RATIONAL  GEOMETRY 

need  of  compasses  for  the  construction  of  a  sect 
product,    althouj:^h   the   compasses   are   n(jt  really 
necessary.     It  seems  that   it  is   not  intended  that 
this  method  be  used  for  the  actual  construction  oi 
the  product  of  sects,  in  case  that  be  required,  the 
definition  being  suited  mainly  to  an  elegant  demon- 
stration of  the  commutative  principle  for  multipli- 
cation  of  sects   without   the  aid   of  Pascal's  the- 
orem.    Were   it   necessary  to  accept   the  truth  of 
Pascal's  theorem  as  given  in  the  "Foundations," 
a    serious    stumbling    block    has    been    met,    and 
Professor  Halsted's  definition   would  be  altogether 
desirable.     All  that  is  required  of  Pascal's  theorem 
for  this  discussion    is   the  special  case  where  the 
two  lines  are  perpendicular,  and  with  this  proved, 
in  the  simple   manner  as   presented    in  this  book, 
using    Hubert's    definition    of    multiplication,  the 
commutative   principle   is   easily  proved.     As  the 
author  makes  use  of  Pascal's  theorem  to  establish 
the  associative  principle,  so  he  might  as  well  have 
used    it    to    establish   the  commutatix'e  principle, 
thus  avoiding  his  definition  of  a  product. 

The  great  importance  of  the  chapter  on  sect 
calculus  is  seen  when  its  connection  with  the 
theory    of  proportion  is  considered.     The  propor- 


HALSTED'S  rational  geometry  25 

tion  a  :  b  ::  a'  :  b'  {a,  a' ,  b,  b'  used  for  sects), 
is  defined  as  the  equivalent  of  tlie  sect  equation 
ab'  =a'b,  following  the  treatment  of  the  "Foun- 
dations." The  fundamental  theorem  of  propor- 
tions and  tiieorems  of  similitude  follow  in  a  man- 
ner quite  simple  indeed  as  compared  with  the 
Euclidean  treatment  of  the  same  subject.  It  is  in 
the  chapter  on  Equivalence  that  the  conclusions 
of  the  preceding  two  chapters,  taken  with  as- 
sumptions Ij.^,  II,  IV,  have  perhaps  their  most 
beautiful  application,  in  the  consideration  of  areas. 
This  subject  has  been  treated  without  the  aid  of 
the  Archimedes  assumpti(jn,  as  Hilbert  had  shown 
to  be  possible.  Polygons  are  said  to  be  equiva- 
lent if  they  can  be  cut  into  a  fmite  number  of 
triangles  congruent  in  pairs.  They  are  said  to  be 
equivalent  by  completion  if  equivalent  polygons 
can  be  annexed  tn  each  so  that  the  resulting  poly- 
gons so  composed  are  equivalent.  These  two 
definitions  are  quite  distinct  and  seem  necessary 
in  order  to  treat  the  subject  of  equivalence  with- 
out assumption  V,  Three  theorems  (§§  26\,  265, 
266)  fundamental  for  the  treatment  are  quite 
easily  proved,  but  the  theorem  Euclid  I,  39,  if 
two  triangles  equivalent  by  completion  have  equal 


26  HALSTED'S  RATIONAL  GEOMETRY 

bases  then  they  have  equal  altitudes,  while  not 
difficult  of  proof,  requires  the  introduction  of  the 
idea  of  area.  The  author  points  out  that  the 
equality  of  polygons  as  to  content  is  a  construct- 
ible  idea  with  nothing  new  about  it  but  a  defini- 
tion. It  is  then  shown  that  the  product  of  alti- 
tude and  base  of  a  given  triangle  is  independent 
of  the  side  chosen  as  base.  The  area  is  defined 
as  half  this  product.  With  the  aid  of  the  dis- 
tributive law  it  is  then  shown  that  a  division  of 
the  triangle  into  two  triangles  by  drawing  a  line 
from  a  vertex  to  base,  called  a  transversal  parti- 
tion, gives  two  triangles  whose  sum  is  equivalent 
to  the  given  triangle.  This  aids  directly  in  the 
proof  of  the  theorem, — if  any  triangle  is  in  any 
way  cut  by  straights  into  a  certain  finite  number 
of  triangles  \  then  is  the  area  of  the  triangle 
equal  to  the  sum  of  the  areas  of  the  triangles  A, 
This  theorem  in  turn  aids  in  the  proof  of  a  more 
general  one  (§  281),  viz.,  if  any  polygon  be  parti- 
tioned into  triangles  in  any  two  different  ways,  the 
sum  of  the  areas  A^  of  the  first  partition  is  the  same 
as  the  sum  of  the  areas  A^^  of  the  second  and 
hence  independent  of  the  method  of  cutting  the 
polygon  into  triangles.     As  the  author  says,  this 


HALSTED'S  RATIONAL  GEOMETRY  27 


is  the  kernel,  the  essence  of  the  whole  investiga- 
tion. It  deserves  complete  mastery  as  it  facili- 
tates the  understanding  of  a  corresponding  theo- 
rem in  connection  with  volumes.  The  area  of  a 
polygyn  is  defined  as  the  sum  of  areas  of  tri- 
angles ^^  into  which  it  may  be  divided,  whence 
it  follows  as  an  easy  corollary  that  equivalent 
polygons  have  equal  area.  The  proof  of  Euclid 
I,  39  is  then  given  with  other  theorems  concern- 
ing area. 

The  mensuration  of  the  circle  discussed  in  this 
chapter,  beginning  with  §  312,  Dehn  character- 
izes* as  an  "energischen  Widerspruch."  It  does 
not  so  impress  the  present  writer.  The  author 
does  not  claim  that  the  sect  which  he  calls  the 
length  of  an  arc  is  uniquely  determined.  It  is 
defined  in  terms  of  betweenness — not  greater  than- 
the  sum  of  certain  sects  and  not  less  than  the 
chord  of  the  arc.  Even  with  a  continuity  as- 
sumption it  cannot  be  uniquely  determined.  But 
the  question  as  to  whether  the  sect  can  be  deter- 
mined uniquely  or  not  can  well  be  left,  as  the 
author  leaves  it,  for  the  one  student  in  ten  thou- 
sand who    may  wish    to    investigate  tt  while  the 

*L.  c,  p.  593. 


28  HALSTED'S  RATIONAL  GEOMETRY 

others  are  occupying  their  time  at  wiiat  may  be  to 
them  a  more  profitable  exercise.  The  definition 
of  the  area  of  a  sector  (§  323),  as  Dehn  says,  * 
"Sielit  im  ersten  Augenblicke  noch  sclilimmer  aiis 
als  sie  in  Wirklichkeit  ist."  Plane  area  has  thus 
far  been  expressed  as  proportional  to  the  product 
of  two  sects.  The  author  could  well  choose  the 
area  of  the  sector  as  k  r  (length  of  arc)  and, 
taking  the  sector  very  small,  the  arc  and  length  of 
arc  may  be  considered  as  one,  in  which  case 
k  /'(length  of  arc)  becomes  the  area  of  a  triangle 
with  base  equal  to  length  of  arc,  and  altitude  r, 
whence  k  =  K  We  then  have  the  sector  area 
defined  in  terms  of  betweenness,  since  the  arc 
length  which  is  included  in  this  definition  was 
thus  defined.  What  geometry  comes  nearer  than 
this,  admitting  all  continuity  assumptions?  In 
any  case  it  can  be  but  an  approximation  and  the 
author  assumes  this. 

The  geometry  of  planes  is  next  considered,  in 
Chapter  XI,  and  the  author  passes  to  a  considera- 
tion of  polyhedrons  and  volumes  in  Chapter  XII. 
The  product  of  the    base  and   altitude  of  a  tetra- 

*  L.  c,  p.  594. 


HALSTED'S  RATIONAL  GEOMETRY  29 

hedron  having  been  shown  to  be  the  same  regard- 
less of  the  base  chosen,  the  tetrahedron  is  made 
to  play  the  same  role  in  the  consideration  of  vol- 
umes that  the  triangle  did  in  the  treatment  of 
areas.  Its  volume  is  defined  as  s-  the  product  of 
base  and  altitude.  The  partitioning  of  the  tetra- 
hedron analogous  to  the  partitioning  of  the  tri- 
angle discussed  in  a  previous  chapter  is  employed 
to  prove  another  "kernel"  theorem,  namely,  if  a 
tetrahedron  7  is  in  any  way  cut  into  a  finite  num- 
ber of  tetrahedra  T^  then  is  always  the  volume 
of  the  tetrahedron  T  equal  to  the  sum  of  the  vol- 
umes of  all  the  tetrahedra  T^.  This  is  one  of  the 
features  of  the  text  as  a  text.  Two  proofs  of  the 
theorem  are  given.  The  second  one,  that  given 
by  D.  O.  Schatunovsky,  of  Odessa,  is  quite 
long.  The  beginner  is  liable  to  get  hopelessly 
swamped  in  reading  it  as  when  reading  some  of 
the  "incommensurable  case"  proofs  of  other 
texts.  He  can  well  omit  It.  The  volume  of  a 
polyhedron  is  defined  as  the  sum  of  the  volumes 
of  any  set  of  tetrahedrons  into  which  it  may  be 
cut.  With  the  introduction  of  the  prismatoid 
formula  and  its  application  to  finding  the  volumes 
of  polyhedrons  we    have    reached    by  easy  steps 


30  HALSTED'S  RATIONAL  GEOMETRY 

another  climactic  point  in  tlie  text.  The  volumes 
of  any  prism,  cuboid  and  cube  follow  as  easy  co- 
rollaries. Contrary  to  the  plan  followed  in  the 
treatment  of  areas,  the  consideration  of  volume  is 
wholly  separated  from  the  consideration  of  equiv- 
alence of  polyhedra.  No  attempt  is  made  to  treat 
the  latter.  If  the  treatment  of  it  be  an  essential 
to  be  considered  in  a  school  geometry  then  a  very 
serious  difficulty  has  been  encountered.  The 
writer  believes  this  is  one  of  a  few  subjects  that 
may  well  be  omitted  from  a  school  geometry.  The 
tendency  has  been,  in  late  years,  too  much  in  the 
other  direction.  Dehn's  criticism*  of  the  proof  of 
Euler's  theorem  (§379)  is  just,  but  it  serves  to 
point  out  but  another  minor  defect  of  the  book. 
In  the  proof  the  terminology  of  motion  is  used 
in  the  statement:  "let  e  vanish  by  the  approach 
of  B  to  y4,"  but  this  is  not  an  essential  method 
of  procedure.  The  demonstration  may  well  be 
begun  thus — if  the  polyhedron  have  but  six  edges 
the  theorem  is  true.  If  it  have  more  than  six 
edges,  then  polyhedra  can  be  constructed  with 
fewer  edges.     Given  a  polyhedron  then  with  an 

*  L.  c,  p.  595- 


HALSTED'S  RATIONAL  GEOMETRY  31 

edge  e  determined  by  vertices  A  and  'B,  construct 
another  with  edges  as  before  excepting  that  those 
for  which  'B  was  one  of  the  two  determining 
points  before  shall  now  \vivq.A  in  its  stead.  Then 
the  new  polyhadron  will  differ  from  the  given  one, 
in  parts,  under  the  exact  conditions  as  stated  in 
the  remainder  of  the  pn^of.  The  restriction  to 
convex  polyhedra,  if  essential,  should  be  made 
clear. 

\x\  the  discussion  of  pure  spherics.  Chapter  XV, 
which  has  to  do  with  the  spherical  triangle  and 
polygon,  we  have  an  excellent  bit  of  non-euclid- 
ean  geometry  whose  results  are  a  part  of  three 
dimensional  euclidean  geometry.  The  plane  is 
replaced  by  the  sphere,  the  straight  by  the  great 
circle  or  straightest,  and  the  planeassumptions  by 
a  new  set  on  association,  betvv^enn^ss  and  con- 
gruences applicable  only  to  the  sphere.  The  pre- 
sentation is  easy  to  comprehend  and  in  fact  much 
of  the  plane  geometry  of  the  triangle  can  be  read 
off  as  pure  spherics.  The  proof  of  the  theorem 
(§  567) — the  sum  of  the  angles  of  a  spherical 
triangle  is  greater  than  two  and  less  than  six  right 
angles — assumes  that  a  spherical  triangle  is  always 
positive.  The  theorem  can  be  proved  in  the  usual 


32         HALSTED'S  RATIONAL  GEOMETRY 

way  by  §  548  and  polar  triangles,  whence  it  fol- 
low^s  as  a  corollary  that  the  spherical  triangle  is 
always  positive,  if  it  be  desirable  to  introduce  the 
notion  of  a  negative  triangle.  In  the  next  and 
last  chapter,  within  the  limits  of  three  pages,  the 
definitions  and  twenty-two  theorems  relating  to 
polyhedral  angles  are  given.  All  these  follow  so 
directly  from  the  conclusions  on  pure  spherics  that 
the  formal  proofs  are  unnecessary.  One  of 
our  widely  used  school  geometries  devotes  as 
many  pages  to  the  definitions  and  a  single  theo- 
rem. This  furnishes  a  sample  of  many  excel- 
lencies of  arrangement  in  the  text. 

While  the  study  of  the  foundations  of  geometry 
has  been,  during  the  last  century,  afield  of  study 
bearing  the  richest  fruitage  for  the  specialist  in  that 
line,  the  results  of  the  study  have  not  hitherto 
served  the  beginner  in  the  study  of  demonstrative 
geometry.  It  seems,  however,  the  day  is  at  iiand 
when  we  can  no  longer  speak  thus.  With  the 
book  before  us.  and  others  that  will  follow,  we  are 
about  to  witness,  it  is  hoped,  another  of  those 
important  events  in  the  history  of  science  whereby 
what  one  day  seems  to  be  the  purest  science  may 
become  the  next  a  most  important  piece  of  applied 


HALSTED'S  RATIONAL  GEOMETRY  33 


science.  Such  events  enable  us  to  see  with  Pres- 
ident Jordan  *  that  pure  science  and  utilitarian 
science  are  one  and  the  same  thing. 

Commendable  features  of  the  text  are,  a  good 
index,  an  excellent  arrangement  for  reference, 
brevity  in  statement,  the  treatment  of  proportion, 
areas,  equivalence,  volumes,  a  good  set  of  original 
exercises,  and  the  absence  of  the  theory  of  limits 
and  "incommensurable  case"  proofs. 

S.  C.  Davisson. 

INDIANA  UNIVERSITY, 
January,  1905. 
[From    the    Bulletin   of    the   American    Mathematical 
Society,  2d  Series,  Vol.  XL,  No.  6,  pp.  330-336.] 


*  Popular  Science    Monthly,   vol.  66,  no.  i,  p.  8i  (No- 
bember,  1904). 


34  HALSTED'S  RATIONAL  GEOMETRY 


Rational  Geometry.  By  George  Bruce  Hal- 
sted,  A.  B.,  A.  M.  (Princeton),  Ph.  D.  (Johns 
Hopkins).  Price  ^1.75.  Chapman  &  HaH. 
Although  so  many  books  on  elementary  geome- 
try are  continually  appearing,  no  apology  need  he 
offered  for  the  publication  of  the  present  work. 
It  has  nothing  in  common  with  the  ordinary 
text-book,  except  that  it  deals  with  the  same  sub- 
ject. Prof.  Halsted  yields  to  none  in  his  rev- 
erence for  the  marvellous  work  achieved  by 
Euclid;  nevertheless,  he  belongs  to  that  school  of 
mathematicians  which  maintains  that  Euclid's 
system  is  not  infallible;  that  his  theory  is,  in  fact, 
built  up  from  an  imperfect  and  incomplete  set  of 
fundamental  axioms  to  which  he  himself  tacitly 
and,  perhaps  even  unconsciously,  added.  In  the 
opinion  of  Prof.  Halsted  and  kindred  thinkers  it 
has  become  necessary,  for  the  advancement  of 
truth,  that  the  system  which  has  held  sole  sway 
for  so  many  centuries  should  give  place  to  another 
and  a  better  one.  Unlike  many  of  the  writers 
who  undertake  the  task  of  reforming  Euclid,  Prof. 
Halsted  shows  no  tendency  to  be  content  with  less 


HALSTED'S  RATIONAL  GEOMETRY  35 

rigid  proof:  on  tlie  contrary,  he    urges  the  neces- 
sity  for   the  utmost  rigour;  and  this,  we  venture 
to    think,  is    one    of  the    strongest    of  his   many 
strong   claims    to    consideration.     He  asserts  that 
the  principles  which  form  the   groundwork  of  his 
book  secure  both  greater  simplicity  and  increased 
rigour  for  his    demonstrations.     Hilhert's  "Foun- 
dations of  Geometry"  furnish    the    basis  for  the 
present  treatise.     Accustomed    as  we  are  to  the 
small  number  of  simply  worded  axioms  which  are 
met  with  in  Euclid,  it  is  somewhat  difficult  to  ac- 
quire   readily   a    comprehensive  grasp  of  the  five 
groups  of  "assumptions"  considered  essential  by 
Hilbert,  and,  seeing  that  an  authority  as  notable 
as  Poincare  failed  to  detect  the  redundancy  of  one 
of     Hubert's     "betweenness    assumptions,"    no 
humbler  mathematician    need    hesitate  to  reserve 
for  a  time   any  definite  expression   of  opinion  as 
to  the  extent  to  which    Hilbert's  "assumptions" 
are  deserving  of  being  regarded  as  unimpeachable. 
Mone,  however,  will  dispute  the  care  and  the  effort 
to  attain  perfection  which  mark    the    drawing  up, 
the  classification,  and  the  enunciation  of  the  "as- 
sumptions"; none  can    fail  to    recognize   how  in 
Prof.  Halsted's    hands  they  yield  simple  and  de- 


36         HALSTED'S  RATIONAL  GEOMETRY 

lightful  proofs  of  many  of  the  propositions  with 
which  every  student  of  matliematics  is  familiar. 
Four  only  of  the  five  groups  of  "assumptions" 
are  used  in  the  present  work,  viz.,  those  in  which 
the  ideas  of  "association,"  of  "betweenness," 
of  "congruence,"  and  of  parallelism  claim  atten- 
tion. The  Archimedean  principle  of  continuity  is 
avoided  in  demonstrating  the  theory  of  proportion, 
and  in  its  place  stands  a  sect  calculus  which  fur- 
nishes for  geometry  an  analogue  to  the  operations 
of  algebra  as  applied  to  real  numbers.  The  asso- 
ciative, commutative,  and  distributive  laws  which 
govern  algebra  are  shown  to  apply  equally  to  the 
sect  calculus  for  geometry.  The  charm  of  many 
of  the  author's  methods  of  proof  has  been  re- 
ferred to:  it  exists  in  a  marked  degree  in  the  sixth 
chapter,  where  the  originality  displayed  in  the 
solution  of  problems  is  specially  attractive.  When 
Hubert's  "Foundations  of  Geometry"  appeared 
there  at  once  arose  in  the  mind  a  doubt  as  to  the 
possibility — at  any  rate,  at  the  present  time — of 
adapting  the  system  to  the  needs  of  the  immature 
student;  but  the  production  of  Prof.  Halsted's 
work  shows  that  no  cause  for  the  doubt  really 
existed. 

[From  The  Educational  Times,  December  i,  1904.] 


HALSTED'S  RATIONAL  GEOMETRY  37 


Rational  Geometry,  based  on  Hilhert's  Foun- 
dations. By  G.  B.  Halsted.  New  York:  John 
Wiley  &  Sons,  1904,  pp.  285. 

We  could  have  wished  that  Mr.  Halsted's 
plan  had  included  a  commentary;  the  matter  is 
<set  out  with  Euclidean  severity. 

Hubert's  first  quarrel  with  the  traditional  geom- 
etry is  about  congruence.  When  is  one  finite 
straight  line  AB  (which  Mr.  Halsted  calls  a 
"sect")  to  be  considered  congruent  with  another 
sect  XY?  Euclid  answers:  When  AB  can  be 
moved  so  as  to  coincide  with  XY.  But,  of  course, 
AB  must  not  alter  in  length  while  it  is  being 
moved.  Now,  what  does  this  mean?  It  means 
that  if  A'B'  is  any  position  of  AB  during  the 
translation,  then  A'B'  is  to  be  congruent  with  AB. 
But  what  does  congruent  mean?  This  is  just 
what  we  are  trying  to  define.  And  we  are  arguing 
in  a  circle.  "To  try  to  prove  the  congruence  as- 
sumptions and  theorems  with  the  help  of  the  mo- 
tion idea  is  false  and  fallacious,  since  the  intuition 
of  rigid   motion  involves,  contains,  and    uses  the 


38  HALSTED'S  RATIONAL  GEOMETRY 

congruence  idea."  To  define  congruence  of  sects 
and  angles  without  motion,  Hilbert  resorts  to  a  set 
of  assumptions.  It  is  curious  tliat  lie  is  forced  to 
assume  Euclid  1.  4  as  far  as  the  equality  of  the 
base  angles:  he  can  then  prove  the  equality  of 
the  bases. 

He  is  unable  to  prove  the  congruence  of  tri- 
angles which  have  congruent  two  pairs  of  angles, 
and  a  pair  of  sides  not  included  (Euclic  1.  26, 
Case  2).  This  appears  to  lead  to  a  second  am- 
biguous case,  as  would  happen  in  the  surface  of  a 
sphere. 

We  learn  that  "no  assumption  about  parallels 
is  necessary  for  the  establishment  of  the  facts  of 
congruence  or  motion."  Playfair's  axiom  is 
adopted. 

Tile  chapter  on  "constructions"  is  interesting. 
Apparently  all  figures  whose  existence  can  be  de- 
duced from  assumptions  admit  of  construction 
with  ruler  and  "sect-carrier,"  c.q.  trisection  of 
sect  is  possible,  and  trisection  of  angle  impossible. 
Hilbert  shovvs  that  there  are  constructions  possible 
with  ruler  and  compass  which  are  not  possible 
with  ruler  and  sect-carrier. 

Coming  to  area,  we   find   the  rejection  of  intui- 


HALSTED'S  RATIONAL  GEOMETRY         39 

tion  leads  as  along  a  thorny  path.  For  reasons 
which  we  dimly  apprehend,  Mr.  Halsted  refuses 
to  associate  numbers  with  sects  (he  never  gives  a 
numerical  measure  of  the  length  of  a  line),  and 
will  have  nothing  to  do  with  limits.  (Hilbert  is 
more  generous.)  Two  polygons  are  defined  as 
equivalent  if  they  can  be  cut  into  a  finite  number 
of  triangles  congruent  in  pairs.  After  proving  the 
equivalence  of  parallelograms  on  the  same  base 
and  between  the  same  parallels,  Hilbert  is  seized 
with  misgivings — perhaps  all  polygons  are  equiv- 
alent. These  doubts  are  resolved,  and  the  section 
ends  with  the  demonstration  that  "a  polygon 
lying  wholly  within  another  polygon  must  always 
be  of  lesser  content  than  the  latter." 

A  similar  procedure  is  necessary  in  dealing  with 
the  volumes  of  polyhedra.  The  area  of  a  sector 
of  a  circle  is  defined  as  the  product  of  the  length 
of  its  arc  by  half  the  radius.  Product  is  defined 
satisfactorily,  and  Mr.  Halsted  lias  a  right  to  define 
"area  of  sector"  as  he  likes;  but  this  definition 
gives  no  clue  to  what  would  be  meant  by  the  area 
of  an  ellipse,  say.  No  general  definition  is  given 
of  the  area  of  a  curved  surface;  but  in  §  453  we 
are  told  that  the  lateral  area  of  a  right  circular  cone 


40         HALSTED'S  RATIONAL  GEOMETRY 

IS  the  same  as  that  of  a  sector  of  a  circle  with  the 
slant  height  as  radius  and  an  arc  equal  in  length 
to  the  length  of  the  cone's  base.  Is  this  a  latent 
definition?  Again,  the  area  of  a  sphere  is  defined 
as  the  quotient  of  its  volume  by  one-third  its 
radius. 

The  volume  of  a  sphere  (or  other  curved  sur- 
face) is  virtually  defined  by  Cavalieri's  assump- 
tion: "If  the  two  sections  made  in  two  solids  be- 
tween two  parallel  planes  by  any  parallel  plane 
are  of  equal  area,  then  the  solids  are  of  equal 
volume."  A  sphere  is  then  compared  in  an  in- 
genious way  with  a  tetrahedron. 

C.  Godfrey. 

WINCHESTER  COLLEGE, 
England. 
[The  Mathematical  Gazette,  Vol.  Ill,  pp.  180-182.] 


HALSTED'S  RATIONAL  GEOMETRY  41 


Rational  Geometry.  By  Prof.  George  Bruce 
Halsted.  New  York:  John  Wiley  &  Sons, 
publishers.  London:  Chapman  &  Hall,  Limited. 
The  modern  standpoint  permits  many  simplifi- 
cations in  the  development  of  geometrical  theory, 
of  which  our  author  skillfully  avails  himself.  Of 
the  many  notable  features  of  this  book  it  suffices 
to  mention  only  the  treatment  of  Proportion, 
Equivalence,  Areas,  Volumes,  Pure  Spherics,  the 
absence  of  the  theory  of  limits,  of  a  continuity 
assumption,  the  presence  of  the  ruler  as  a  sect- 
carrier  displacing  the  compasses.  This  volume  of 
285  pages  contains  all  that  is  essential  to  a  course 
in  elementary  geometry.  The  language  is  simple, 
the  logic  exact,  the  exposition  masterly,  as  was  to 
be  expected  from  Dr.  Halsted.  The  book  seems 
admirably  adapted  to  class  use.  The  already 
great  indebtedness  of  teachers  of  geometry  to  Dr. 
Halsted  has  been  manifoldly  increased  by  the  pub- 
lication of  this  book,  which,  in  the  opinion  of  the 
writer  and  with  no  intended  disparagement  of 
others,  is  the    most  important  contribution  to  the 


42  HALSTED'S  RATIONAL  GEOMETRY 

text-book  literature  of  elementary  geometry  that 
has  appeared.  And  now  that  the  way  has  been 
opened  may  we  not  hope  that  the  teachers  of  ge- 
ometry in  the  secondary  schools  and  colleges  will 
see  to  it  that  the  present  generation  of  pupils  shall 
receive  the  benefits  rightly  accruing  to  them 
through  the  profound  researches  of  the  present 
and  last  centuries  on  the  foundations  of  geometry. 

T.  E.  McKlNNEY. 

MARIETTA,  O. 

[From    the    review    in    The    American    Mathematical 
Monthly.] 


HALSTED'S  RATIONAL  GEOMETRY  43 


HALSTED'S  GEOMETRY   IN  HINDUSTAN. 

In  a  leader  in  "Indian  Engineering"  (Published  at  7, 
Government  Place,  CalcDtta),  the  editor,  praising  Hal- 
sted's  Elements  of  Geometry,  had  said  : 

The  elements  of  old  immortal  Euclid  have  been  used  as 
THE  text-book  on  the  subject  of  geometry  for  twenty-two 
centuries  in  all  countries  of  the  modern  world  which 
derive  their  culture  and  civilization  from  the  Greek  ;  in- 
deed so  close  has  been  the  association  of  Euclid  with 
geometry,  that  not  unnaturally  the  name  of  Euclid  is 
used  in  common  parlance  as  synonymous  with  the  science 
of  geometry.  But  though  he  has  worn  the  crown  so  well 
and  so  long,  within  the  last  century  the  foundations  of 
the  science  have  been  examined  anew  by  Iha  mighty 
intellects  of  Lobachevsky,  Bolyai,  Riemann,  and  others — 
men  worthy  of  a  seat  by  the  side  of  Archimedes  and  New- 
ton ;  and  the  penetrative  insight  of  men  like  these  has 
shown  that  the  vision  of  Euclid  was  limited,  that  the 
boundaries  of  the  science  are  not  where  he  placed  them, 
that  the  system  he  reared  on  the  basis  of  the  so-called 
twelfth  axiom  is  not  one  of  the  necessities  of  the  human 
intellect,  and  that  it  is  quite  possible  to  construct  a  con- 
siscent  system  of  geometry  in  which  both  the  twelfth 
axiom  and  the  thirty-second  proposition  of  the  first  book 
of  Euclid  are  violated.  Dr.  Halsted  has  been  one  of  the 
foremost  captains  in  the  work  of  popularizing  the  re- 
searches of  the  investigators  we  have  named,  and  has 
thus  materially  facilitated  the  exploration  of  the  new 
country.     We  have  always  regretted  that  these  beautiful 


44  HALSTED'S  RATIONAL  GEOMETRY 


researches,  so  stimulating  and  fascinating  to  the  imagina- 
tion, are  not  presented  in  a  form  in  which  they  can  be 
readily  assimilated  by  the  beginner,  and  we  venture  to 
hope  that  Dr.  Halsted,  who  is  so  well  qualified  for  the 
task,  will  deal  with  the  subject  definitely  in  a  companion 
volume  to  the  work  now  before  us. 

[What  the  learned  editor  ventured  to  hope  has  come  now 
to  fruition,  as  signaled  by  the  following  review  in  "Indian 
Engineering,"  Vol.  XXXVll,  No.  22,  June  3,  igc;,  by 
Wm,  John  Greenstreet,  F.  R.  A.  S.,  editor  of  the  "Math- 
ematical Gazette,"  the  official  organ  of  the  British  Asso- 
ciation for  the  ImprovemenrGeometrical  Teaching:] 


RATIONAL  GEOMETRY 

Under  the  above  name  Professor  G.  B.  Halsted 
has  published  a  volume  which  is  sure  to  attract 
attention  from  those  who  have  followed  the  work 
that  has  been  accomplished  by  Hilbert  in  the 
study  of  the  foundations  of  geometry.  The  book 
before  us  is  certain  to  attract  more  than  ordinary 
attention,  being  the  first  essay  in  the  introduction 
of  the  new  ideas  into  the  teaching  of  ele- 
mentary geometry.  The  author  is,  of  course, 
well  known  to  mathematicians  all  over  the  world, 
being  the  most  doughty  and  intrepid  advocate  of 
general  geometry  in  the  United  States.  Time 
alone   will    show    whether  the  present  effort  will 


HALSTED'S  RATIONAL  GEOMETRY         45 


command    more   than    a   sneers    d'estime.      One 
wonders  whether  the   American  teacher  will  over- 
come all  his  prejudices  and  set  to    work  to  master 
the  new  ideas  so  ably  herein  set  forth.    To  many, 
no  doubt,  the  difficulties  will  be    repellent,  and  if 
that    be   so,  when    the   tide   turns,    and   general 
opinion  is  ripe  for  the  adoption   of  the  new  ideas, 
the    recalcitrants    will    have    to    be    "mended  or 
ended."     The   change  will  not  be  as   pleasantly 
made  as    was  the  case  when  the  proposals  of  the 
Mathematical    Association    were   adopted    by  the 
universities   and    teaching   bodies  in    Britain,  for 
British  opinion  had  long  been  ripe  for  the  change. 
So   far   as    we  can  judge,  Americans  have  as  yet 
exhibited    but   a  mild  curiosity  as  to  the  scope  of 
the  changes    in    the  teaching  of  geometry  in  the 
old  country.     The  book  before  us  makes  a  much 
more  serious  demand  on  the  patience   and  the  in- 
tellect  of  the   teacher,  and  one  wonders  whether 
the  cheque  will  be  honoured   until  after  consider- 
able preliminary  delay.     For  this  volume  marks  a 
tremendous    breach    with    the   traditions    of   two 
thousand    years.     It   sounds    the    death  knell  of 
intuition,    and   at   first   one   can    hardly  think  of 


46         HALSTED'S  RATIONAL  GEOMETRY 

geometry  without  intuition.  11  y  a  plus  de  qiiar- 
ante  ans  que  jc  dis  de  la  prose  sans  que  j'en  susse 
Hen!  said  M.  Jourdain  in  Moleire's  Bourgeois 
Gentilhomme.  And  a  little  consideration  will 
show  how  often  our  work  and  our  methods  in 
geometry  have  been  unconsciously  intuitional. 
Another  point  which  will  militate  in  some  measure 
against  the  success  of  this  book  in  so  conserva- 
tive a  land  as  Britain  is  the  predilection  of  Pro- 
fessor Halsted  to  adopt  novelties  of  nomenclature. 
We  do  not  mean  but  that  in  most  cases  he  may 
be  able  to  advance  sufficient  justification  for  a 
course  which  always  has  great  drawback's,  and 
especially  when  the  change  concerns  words  which 
are  wrought  into  the  warp  and  woof  of  the  lan- 
guage. Sometimes  the  change  happens  to  be 
both  timely  and  happy.  When  a  word  has  more 
than  one  connotation  it  is  time  that  it  disappeared. 
For  this  reason  it  is  high  time  that  "axiom" 
should  be  relegated  to  the  limbo  of  words  that 
have  outlived  their  use,  and  we  cannot  object  to 
the  ingenious  substitute — assumption .  So  again, 
the  word  "segment"  in  "segment  of  a  line"  and 
"segment  of  a  circle"  is   at  times,    and  to  a  cer- 


HALSTED'S  RATIONAL  GEOMETRY         47 

tain  order  of  mind,  provocative  of  confusion. 
Segment  is  retained  for  the  circle,  but  tlie  segment 
•of  a  line  is  called  by  the  author  a  "sect,"  the 
instrument  for  the  transfer  of  segments  (streck- 
eniibertrager)  being  a  "sect-carrier."  "Co- 
punctal"  is  hideous,  but  then  it  has  a  great  advan- 
tage over  "concurrent,"  first  because  the  latter 
involves  the  idea  of  motion,  and  secondly  because 
the  word  co-punctal  expresses  exactly  what  is 
intended,  i.e.,  the  possession  of  a  common  point. 
But  we  shudder  at  co-straight  in  place  of  co-linear. 
Hilbert's  Second  Group  of  Axioms,  we  beg  par- 
don— assumptions,  defined  the  idea  expressed  by 
"between,"  and  were  called  axioms  of  order. 
Professor  Halsted  calls  them  "betweenness  as- 
sumptions," to  which  there  is  no  objection. 
Chapters  1-IV  state  the  assumptions,  and  give  a 
few  theorems  which  miy  be  deduced  from  them. 
The  assumptions  are  divided  by  Hilbert  into  five 
groups: — connection,  order  or  betweenness,  par- 
allels (Euclid's),  congruence,  continuity  (Ar- 
chimedes'). The  order  is  logical  enough.  First 
the  blade,  then  the  ear  and  then  the  full  corn  in 
the  ear.  First  the  definition  of  the  geometrical 
entities — point,  line,    plane,  space;    then    the  as- 


48  HALSTED'S  RATIONAL  GEOMETRY 

sumptions  which  are  made  as  to  the  mutual  rela- 
tions of  the  entities.  The  assumptions  of  con- 
nection are  seven;  by  tlieir  means  we  can  show- 
that  two  co-planar  straight  lines,  "straights," 
cannot  have  two  points  in  common;  they  must 
have  one  common  point  or  none,  with  similar 
properties  of  planes.  Next  we  have  the  four  as- 
sumptions of  betweenness,  first  treated  properly 
by  W.  Pasch.  Hilbert  originally  gave  five,  buf 
the  fourth  was  shown  by  R.  L.  Moore  to  be  in- 
cluded in  the  others.  The  last  of  these  assump- 
tions will  show  the  reader  the  extent  to  which  we 
are  left  independent  of  intuition.  Draw  a  triangle 
ABC.  Any  co-planar  line  which  cuts  AB  will 
also  cut  either  BC  or  AC.  That  is  now  an  as- 
sumption! The  general  theorem  of  congruence  is 
as  follows: — If  ABC A'B'C  are  two  con- 
gruent figures  and  P  any  point  in  the  first  we  can 
always  find  a  point  P'  in  the  second  such  that  the 
figures  ABC  P,  A'B'C  P'  are  congruent.  This 
brings  us  to  the  idea  of  displacement,  which  is 
logically  dependent  on  that  of  congruence.  The 
last  of  the  four  chapters  contains  a  signal  instance 
of  the  fading  glories  of  intuition,  tor  the  author 
proves  that  every  straight   line  has  a  middle  point 


I 


HALSTED'S  RATIONAL  GEOMETRY  49 

and  that  every  angle  has  a  bisector!  We  next 
have  the  chapter  on  the  circle  and  its  properties. 
We  may  point  out  an  e.xcellent  innovation — the 
word  circumference  disappears.  If  C  be  any 
point  in  a  plane,  the  aggregate  of  all  the  points  A 
in  the  plane  for  which  tlie  sects  CA  are  congruent 
to  one  another  is  a  circle.  At  the  end  of  this 
chapter  we  come  to  Archimedes'  assumption, 
which  has  not  yet  been  used.  We  must  not 
omit  to  mention  that  Professor  Halsted  missed  an 
opportunity  of  improving  the  proof  of  the  theorem 
that  the  plane  is  divided  into  two  parts  by  a  poly- 
gon. The  proof  as  given  has  been  shown  by 
Dehn  to  hold  good  in  the  case  when  the  polygon 
is  a  triangle,  but  not  otherwise.  Next  comes  con- 
structions. Whenever  a  construction  is  dependent 
on  theorems  based  on  the  assumptions  they  re- 
quire for  their  solution,  only  the  straight-edge  and 
the  sect-carrier  are  necessary,  and  thus  they  in- 
volve only  the  drawing  of  a  line  and  the  cutting 
off  on  it  a  given  sect.  Chapter  VIII  is  devoted 
to  what  used  to  be  called  the  algebra  of  segments, 
but  is  now  the  "sect-calculus."  Proportion  and 
Similitude  form  the  subject  matter  of  Chapter  IX, 
and  Chapter    X  deals  with  areas.     In  the  twelfth 


GEORGE  BRUCE  HALSTED 


HALSTED'S  RATIONAL  GEOMETRY  53 


GEORGE  BRUCE  HALSTED 

La  Societa  Americana  pel  progresso  delle  Scienze 
nel  suo  50°  congresso  tenuto  a  Pittsbourgh  dal  28 
giugno  al  3  luglio  1902  ha  eletto  Presidente  della 
sezione  "  Matematiche  ed  Astronomia  "  il  Pro- 
fessore  GEORGE  BRUCE  HalSTED.  Pigliando 
occasione  dalla  lusinghiera  e  meritata  distinzione 
ottenuta  dall'egregio  nostra  collaboratore  (1)  ed 
amico,  d  permettiamo  dire  qualclie  parola  di  Lui. 

II  Prof.  G.  B.  HaLsted  Iia  avuto  dalla  natura 
il  dono  poco  comune  di  poter  accoppiare  nel  modo 
piu  simpatico  grande  modestia  e  grande  bonta 
d'animo  ad  un'  erudizione  estesissima,  tanto  da 
far  stare  dubbioso  chi  lo  avvicina    se    in  lui  debba 

(i)  Oltre  all'aver  fatto  conoscere  negli  Stati  Unit!  la 
presente  Rivista  con  parole  di  simpatia  egli  tradusse  per 
due  delle  niu  diffuse  Riviste  scientifiche  americane. 
rAmerican  Mathematical  Monthly  e  il  Science,  N.  S.,  le 
note  scritte  nel  nostro  periodico  da  Juan  J.  Duran  Loriga 
(Charles  Hermite,  vol.  I,  pag.  2)  e  da  P.  Barbarin 
(Sull'utilita  di  studiare  la  Geometria  non-euclidea,  vol.  I, 
pag.  85.) 


54         HALSTED'S  RATIONAL  GEOMETRY 

piu  ammirare  le  doti  della  mente  o  quelle  del 
cuore.  E  pero  certo  che  chi  relazione  con  lui  e 
tratto  ad  affezionarglisi  sinceramente. 

Nato  il  25  novembre  1853  da  una  famiglia  di 
studios!  che  prese  larga  parte  alia  rivoluzione 
americana,  ha  11  vanto  di  essere  diretto  discen- 
dente  di  queirAbramo  Clark  che  fu  il  firmatario 
della  dichiarazione  d'  indipendenza.  Comincio 
giovanissimo  a  mostrare  la  sua  predilezione  per  lo 
studio  del  scienze  esatte  col  distinguersi  in  esse  e 
meritarsi  continui  premi  nelle  classi  successiva- 
mente  frequentate,  benche  varie  ore  egli  dovesse 
sottrarre  agli  studi  per  guadagnarsi  i  mezzi  di  con- 
tinuarli.  Ma  le  sue  doti  di  studioso  furono  ancor 
piu  apprezzate  al  suo  ingresso  nella  carriera  dell'in- 
segnamento  e  gli  valsero  un  pubblico  elogio  dell'il- 
lustre  Prof.  Sylvester  ed  una  calorosa  raccomand- 
azione  di  questi  al  Prof.  Borchardt  quando  il  gio- 
vane  Halsted  si  reco  in  Germania  a  compiervi  un 
corso  di  perfezionamento.  Piu  tardi  fu  lo  stesso 
Sylvester  a  presentarlo  alia  Societa  Matematica 
di  Londra.  Nel  1879  ricevetti  il  grado  di  Dottore 
in  Filosofia  neH'Universita  Johns  Hopkins,  e  dopo 
aver    inaugurato    ed   anche    diretto    per   qualche 


HALSTED'S  RATIONAL  GEOMETRY         55 


tempo  "Post  Graduate  Instuction"  in  Princeton. 
La  sua  opera  di  scienziato  fu  opera  feconda  e  si 
esplico  in  campi  diversi  rendendone  il  sui  nome 
popolarissimo  in  ogni  categoria  di  studiosi.  Negli 
otto  anni  passat  all'Universita  John  Hopkins  pub- 
blico  in  diverse  riviste  scientifiche  una  lunga  serie 
di  note  e  memorie  sui  quaternioni,  sui  determin- 
anti,  sulla  storia  delle  matematiche,  sull'algebra 
moderna:  pubblico  inoltre  un 'opera  didattica,  di- 
venuta  classica,  la  Geometria  metrica  (1),  ove, 
con  principi  e  metodi  nuovi  e  esposto  quanto  si 
riferisce  alia  misura  delle  lunghezze,  delle  aree, 
dei  volumi,  degli  angoli.  Quest'opera  fu  tanto 
bene  accolta  no  solo  in  America,  ma  anche  in 
Inghilterra,  da  valergli  I'onore  di  essere  quasi 
integralmente  riportata  da  Wm.  Thomson  nella  9^ 
edizionedeir  "Encyclopaedia  Britannica"  alia  voce 
"Mensuration,"  e  da  far  scrivere  al  venerando 
Sim.  Newcomb:  "Halsted  e  autore  del  tratto  sulla 
misura  che  e  il   migliore  e  il   piu  complete  che  io 


conosca." 


La  grande  operosita  di  Halsted   non  venne  certo 


(i)  Metrical    Geometry,     An    Elementary  Treatise  on 
Mensuration.     Boston:  Ginn  and  Co. 


56  HALSTED'S  RATIONAL  GEOMETRY 

meno;  e  stanno  a  fame  fede  i  numerosi  lavori  che 
portano  il  suo  nome.  Parecchi  di  essi  hanno  per 
scopo  la  volgarizzazione  della  scienza  e  sono  disse- 
minati  nel  Monist,  nella  Educational  Review,  nella 
Popular  Science  Monthly,  nell 'American  Mathe- 
matical Monthly,  ecc. 

Fra  le  sue  opere  didattiche,  (1)  ed  oltre  a  quella 
precedentemente  ricordata,  merita  speciale  atten- 
zione  la  "Elementary  Syntetic  Geometry  (New 
York,  J.  Wiley  and  Sons),  nella  quale  sono  logi- 
camente  riuniti  e  rigorosamente  esposti  i  punti 
principali  della  Geometria  sintetica.  Essa  e  una 
della  piu  rimarchevoli  nella  letteratura  didattica 
americana  e  ne  fa  fede  quanto  di  essa  scrisse  una 
delle  riviste  piu  autorevoli  ed  imparziali:  (2) 
"Per  piu  di  2000  anni  la  Geometria  ebbe  per  fon- 
damento  esclusivo  la  congruenza  dei  triangoli:  si 
presenta  ora  un  libro  che  giunge  ai  risultati  stessi 
senza  fare  uso  alcuno    dei  triangoli    congruenti  e 

(i)  Altre  di  tali  opere  sono  ad  esempio:  Mensuration— 
(Ginn  &  Co,,  Boston  and  London);  Elements  ot  Geome- 
try. (J.  Wiley  and  Sons,  New  York),  Projective  Geom- 
etry.    (Ibidi. 

(2)  Bullet,  of  the  New  York  Mathematical  Society,  t. 
Ill,  N.o  I,  pag.  8-14. 


HALSTED'S  RATIONAL  GEOMETRY  57 

con  tale  semplicita  che,  ad  esempio,  tutti  i  casi 
ordinari  della  coiT^ruenza  dei  triangoli  sono  dimo- 
strati  in  otto  righe." 

11  nome  di  Halsted  e  indissolubilmente  collegato 
alia  volgarizzazione  della  Geometria  non-euclidea. 
(4)     Da  quando  comincio  ad  appassionarsi    agli 

(4)  Hanno  per  scopo  la  volgarizzazione  della  Metage- 
ometria  le  sue  pubblicazioni  : 

Gauss  and  the  Non-Euclidean  Geometry,— Science,  N.S., 
t.  XIV,  pag.  705-717  (1890);  The  Appreciation  of 
Non-Euclidean  Geometry,— ibid.,  pag.  462-465. 

Lambert's  Non-Euclidean  Geometry,— Bui.  of  the  New- 
York  Math.  Society,  t.  Ill,  pag.  78-80,  11894). 

Non-Euclidean  Geometry ;  Historical  and  Expository,— 
American  Math.  Monthly,  t.  I,  H,  III,  (1894,  95,  96). 

The  Non-Euclidean  Geometry  inevitable,— The  Monist, 
t.  IV,  Chicago,  1894. 

Some  salient  points  in  the  History  of  Non-Euciidean  Ge- 
ometry and  Hyper-Spaces,— Math.  Papers  read  at  the 
Internat.  Math.  Congress,— Chicago,  1893. 

Nicolai  1,  Lobatchefsky,— Address  prononced  at  the  com- 
memorative meeting  of  the  Imperial  University  of 
Kasan,  October  22,  1893,  by  A.  Vassilief  (trad,  dal 
russo,  con  prefazione).    Austin,  1894. 

Darwinism  and  Non-Euclidean  Geometry,— Boll,  di  Ka- 
san, (2),  t.  VI,  pag.  22-25,  (1896). 

The  Introduction  to  Lobatchefsky's  new  elements  of 
Geometry,— Texas-Academy,  1897, 

Scientific  Books,  Urkunden  zur  Geschichte,— Science,  N. 
S.  t.  IX,  pag.  813-817,  (1889). 

Report  on  Progress  in  Non-Euclidean  Geometry,— Proc. 
of  the  Amer.  Ass.  f.  adv.  of  Sc.  t.  XLVIII,  pag.53-68 
C1899), 


58  HALSTED'S  RATIONAL  GEOMETRY 

studi  filosofici  commincio  ad  interessarsi  a  questa 
Geometria  per  divenirne  in  breve  non  solo  cultore 
ma  apostolo  entusiasta  guadagnandosi  il  vanto  di 
darne  la  prima  bibliografia,  (1)  cosi  importante 
da  essere  subito  tradotta  e  ristampata  in  Russia. 

Da  pochi  anni  aveva  avuto  principio  in  Europa 
quel  periodo  scientifico  nel  quale  1'  attenzione  del 
geometri  era  stata  richiamata  sulle  ricerche  relative 
ai  fondamenti  della  Geometria  e  su  quella  Geome- 
tria per  la  quale  Sylvester  proponeva  la  denomi- 
nazione  di  iiltra-enclidea.  J.  Hoiiel  in  Francia  e 
I'immortale  Beltrami  in  Italia  erano  quasi  soli  a 
segnalare  I'alta  importanza  dei  lavori  di  Lo- 
batchefsky,  di  Bolyai,  ed  a  fare  intravedere 
come  lo  studio  delle  basi  della  scienza  dovesse 
assorgere  alia  piu  alta  importanza  filosofica,  poten- 

Non-Euclidean  Geometry,— Am.  Math.  Month.,  t.  VII, 
pag.  123-133,  (1900). 

Non-Euclidean  Geometry  for  Teachers,— Popular  As- 
tronomy, 1900. 

Supplementary  Report  on  Non-Euclidean  Geometry,— 
Science,  N.  S.  t.  XIV,  pag.  705-717,  U901). 

The  Teaching  of  Geometry,— Educational  Review,  New 
York,  Dec.   1902,  pag. 456-470. 

(i)  Bibliography  of  Hyperspace  and  Non-Euclidean 
Geometry.— Amer.  Jour,  of  Math.  vol.  I,  pag.  261-266  e 
384-385,  (1878);   vol.  II,  pag.  65-70,  (1879). 


HALSTED'S  RATIONAL  GEOMETRY  59 

do  forse  diventare  I'unico  capace  di  darci  le  chiavi 
delle  origini  e  della  formazione  delle  conoscenze 
umane.  Che  meraviglia  dunque  che  nella  giovane 
America  nessuno  si  fosse  ancor  messo  alia  testa  di 
coloro  che  ambivano  di  essere  ammessi  nella  scir 
ola  che  aveva  mostrato  che  quella  Geometria  che 
per  piu  di  duemila  anni  era  stata  ritenuta  Tunica 
possibile  non  poteva  reggere  ad  una  seria  discuss- 
ione  dei  suoi  postulati  e  che  altri  sistemi  di  Ge- 
ometria, egualmente  rigorosi,  erano  possibili? 

Ivi  pure  piu  di  uno  aveva  cominciato  a  discu- 
tere  le  due  proposizioni  di  Legendre,  la  cui  dimo- 
strazione  implica  I'assioma  d'Archimede,  ed  aveva 
mostrato  che  cosa  poteva  diventare  questa  Geom- 
etria, privata  di  tale  postulato,  e  quella  di  Euclide, 
privata  del  suo  Xl°  assioma.  Tutto  ciu  pero  res- 
tava  nell'esclusivo  dominio  dei  dotti,  anzi  di  quei 
pochi  che  erano  iniziati  ai  nuovi  studi.  Halsted 
si  assunse  I'incarico  di  porre  alia  portata  di  tutti  i 
nuovi  studi,  traducendo  le  opere  del  russo  Lobat- 
chewsky,    (1)  dell'ungherese  Bolyai,    (2)    dell'- 

(i)  N.  Lobatchefsky, — Geometrical  Researches  on  the 
Theory  of  Parallels— (trad,  dal  russo,  con  prefazione  e 
appendice),— Tokyo  Sugakubutsurigiku  Kawai  Kiji,  t. 
V,  pag.  6-50,  (1894). 


60  HALSTED'S  RATIONAL  GEOMETRY 

italiano  Saccheri:  (3)  fu  il  suo  entusiasmo  che 
trascino  molti  nella  via  da  quel  sommi  segnata,  e 
ben  presto  una  bella  schiera  di  nomi  eletti  venne 
ad  arrichire  la  falange  dei  cultori  delle  nuove  idee. 
Postisi  al  corrente  dei  lavori  dei  geometri  non- 
euclidei,  pienamente  iniziati  alia  tradizione  filoso- 
fica,  dominati  da  spirito  critico  di  raro  vigore,  con- 
tribuirono  in  breve  anch'essi  a  porre  in  evidt-nza 
gli  errori  e  i  controsensi  filosofici  dei  metageometri 
ed  a  debellare  le  obiezioni  ingiuste  e  spesso  igno- 
ranti  indirizzate  dai  filosofi  alia  metageometria.  Si 
schierarono  anch'essi  fra  coloro  che  vollero  restau- 
rare  e  correggere  le  teorie   criticiste    mostrandosi 

Id.  — The  Non-Euclidean  Geometry, — Geometrical  Re- 
searches on  the  Theory  of  Parallels,  (trad,  dal  russo), 
Austin,  1894. 

Id.  New  principles  of  Geometry,  with  a  complete 
Theory  of  Parallels,— (trad,  dal  russo),  Austin,  J897. 

(2)  J.  Bolyai,— The  Science  absolute  of  Space,  indepen- 
dent of  the  truth,  etc.  (trad,   dal    latino), — Austin,  1894, 

e  riportato  anche  in  Tokyo  Sugaku ,  t.  V,  pag.  94-135, 

1894. 

(3)  Euclides  ab  omni  naevo  vindicatus,  sive  conatus 
geometricus  quo  stabiliuntur  prima  ipsa  universae  Geo- 
metriae  principia, — Auctore  Hieronymo  Saccherio,  Socie- 
tate  Jesu,  in  Ticinensi  Universitate  Matheseos  Professore 
— Mediolani.     1733. 


HALSTED'S  RATIONAL  GEOMETRY         61 

discepoli  e  continuatori  di  Kant,  sintetizzando  ogni 
anteiiore  ricerca  nella  Teoria  dei  Gruppi  che  per- 
mise  a  Sophus  Lie  di  ridurre  gli  assiomi  della  Ge- 
ometria  alia  loro  logica  essenza. 

E  se  questo  contributo  di  gratitudine  chela  sci- 
enza  deve  ad  Halsted  sia  giusto  valga  a  confer- 
maiio  il  giudizio  che  di  lui  da  I'illustre  Prof.  A. 
Vassilief  dell'Universita  di  Kasan  in  lettera  indi- 
zzatami  in  questi  ultimi  giorni: 
"Nella  stoiia  della  difiusione  delle  idee  della 
Geometria  non-euclidea  il  nome  di  Halsted  sara 
sempre  menzionato  con  grande  stima.  E  state 
lui  a  dare  la  prima  bibliografiia  delle  opere  sulla 
Geometria  non-euclidea;  e  stato  lui  ad  offrire  il 
suo  eminente  appoggio  all'opera  del  Comitate 
Lobatchefsky  fondato  a  Kasan  nel  1893  alio 
scopo  di  celebrare  la  memoria  del  grande  geom- 
etra  russo;  e  stato  lui  a  dare  la  traduzione  in- 
glese  di  varie  opere  di  Lobatchefsky,  ed  e  stato 
ancor  lui  a  far  conoscere  al  pubblico  scientifico 
anglo-sassone,  in  una  serie  d'articoli  sempre  in- 
teressanti,  tutte  le  novita  letterarie  della  Geom- 
etria non-euclidea,  Questo  ardore  instancabile 
col      quale    il    distinto      professore     si    occupa 


62         HALSTED'S  RATIONAL  GEOMETRY 


"di  tutto  cio  che  si  lega  alia  Geometria  non-eu- 
clidea  deriva  daH'interesse  filosofico  e  gnoseo- 
logico  che  essa  offre  per  lui.  Egli  ha  molto  lu- 
cidamente  esposto  questo    interesse  nel  suo  arti- 

"colo  "Darwinism  and  Non-Euclidean  Geometry" 
scritto  a  mia  preghiera  durante  il  suo  soggiorno 
a  Kasan  e  del  quale  conservero  sempre  la  piu 
cara  memoria.  11  lungo  viaggio  dal  Texas  alle 
rive  del  Volga,  fatto  col    solo    intento  di  onorare 

"la  memoria  di  Lobatchefsky  e  anch'esso  prova 
dell'amore,  — e  posso   anche  dire  del  fanatismo, 

" — del  Prof.  Halsted  per  questo  ramo  della  Sci- 
enza  geometrica.      Ma    senza    fanatismo   non  si 

"puo  fare  nulla  di  grande,  e  son  sicuro  che  la 
letteratura  scientifica  americana  ricevera  hen 
presto  da    parte  di    Halsted  una  storia  completa 

"della  Geometrica  non-euclidea,  che  noi  non  pos- 

"sediamo  ancora.     Sara  il  degno  coronamento  dei 

"suoi  sforzi  per  propagare  le  idee  di  Lobatchefsky 

"e  di  Bolyai  nella  letteratura  anglo-americana." 
Ed    e   appunto    cio    che  anch'io  mi  auguro  nel 

porgere    all'egregio     Professore     il     piu     fervido 

augur  io    e  il  piu  affettuoso  saluto. 

Prof.  C.  Alasia. 

TEMPIO  (SARDEGNA), 
Marzo,  1903. 


HALSTED'S  RATIONAL  GEOMETRY  63 


GEORGE  BRUCE  HALSTED 

The  Italian  Biography,  by  Professor  Cristoforo 
Alasia  De  Quesada,  translated  by  Miss  Mar- 
garet A.  Gaffney,  of  Whitman,  Massachusetts. 

The  American  Association  for  the  Advancement 
of  Science,  at  its  50th  meeting,  held  in  Pittsburg 
from  June  28  to  July  3,  1902,  elected  as  president 
of  the  section  for  Mathematics  and  Astronomy, 
Professor  GEORGE  BRUCE  HALSTED.  This  flat- 
tering and  deserved  honor  conferred  upon  our 
distinguished  collaborator  (1)  and  friend  gives  us 
an  opportunity  to  say  a  few  words  about  him. 

Nature  has  bestowed  upon  Professor  Halsted 
the  rare  gift  of  being  able  to  unite  in  the  most  at- 

(i)  Besides  having,  with  sympathetic  words,  made 
this  magazine  known  in  the  United  States,  he  has  trans- 
lated for  two  of  the  American  scientific  journals  of 
widest  circulation,  the  American  Mathematical  Monthly 
and  Science,  N.  S.,  the  articles  written  in  our  periodical 
by  Juan  J.  Duran-Loriga  (Charles  Hermite,  vol.  I, 
pag.  2)  and  by  P.  Barbarin  (SulT  utilita  di  studiare  la 
Geometria  non-euclidea,  vol.  1,  pag.  85). 


64  HALSTED'S  RATIONAL  GEOMETRY 

tractive  manner  great  modesty  and  great  kindness 
of  disposition  to  very  deep  and  extensive  learning, 
so  much  so  as  to  make  all  who  approach  him 
doubt  whether  to  admire  the  more  the  gifts  of  his 
mind  or  of  his  heart,  it  is  certain  that  his  asso- 
ciates come  to  feel  for  him  the  deepest  attach- 
ment. 

Born  November  25,  1853,  of  a  family  of  schol- 
ars that  took  an  important  part  in  the  American 
Revolution,  Professor  Halsted  can  claim  direct 
descent  from  Abram  Clark,  a  signer  of  the 
Declaration  of  Independence.  He  began  when 
very  young  to  show  his  predilection  for  the  study 
of  the  exact  sciences,  distinguishing  himself  in 
these,  and  continually  v\inning  honors  in  his 
successive  classes,  although  he  several  times 
withdrew  from  his  studies  to  secure  the  means  of 
continuing  them.  But  his  gifts  as  a  scholar  were 
even  more  appreciated  when  lie  began  teaching, 
and  young  Halsted  won  a  public  eulogy  from  the 
eminent  Prof.  Sylvester,  and  a  warm  recommen- 
dation from  him  to  Prof.  Borchardt  when  he  went 
to  Germany  to  take  a  finishing  course.  Later, 
Prof.  Sylvester  also  introduced  him  to  the  London 
Mathematical  Society.     In  1879    he    received  the 


HALSTED'S  RATIONAL  GEOMETRY  6S 

degree  of  Doctor  of  Philosophy  from  Johns  Hop- 
kins University.  Shortly  after  he  organized  and 
for  some  time  directed  the  "  Post-Graduate  hi- 
struction  "  at  Princeton. 

His  work  as  a  scientist  was  fertile,  illuminating 
diverse  subjects,  thereby  making  his    name  pop- 
ular among  all  classes  of  students.     \n  the  years 
passed  at  Johns  Hopkins  University  he  published 
in  different  scientific  reviews  a  long  series  of  notes 
and  memoirs  on  quaternions,  on  determinants,  on 
the  history  of  mathematics,  on    modern  algebra. 
He  published  also  his  Metrical  Geometry  [Boston, 
Ginn  &  Co.],  a  text  book  now  become  a  classic. 
In   this  by   new    principles   and   methods  he   ex- 
pounds   what  pertains    to    the    measurement    of 
lengths,  areas,  volumes,  and  angles.     This  work 
was  so    well  received,  not   only  in    America,  but 
in  England,  that  it  had  the   honor  of  being  almost 
entirely  reproduced  by  Wm.  Thomson   in  the  9th 
edition  of  the  Encyclopaedia  Brittanica  under  the 
title,    "Mensuration."     It  caused    the    venerable 
Simon  Newcomb  to  write  of  Dr.  Halsted  :     "He 
is  the  author    of  a  treatise  on  Mensuration  which 
is  the  most  thorough  and    scientific  with  which  I 
am  acquainted." 


66  HALSTED'S  RATIONAL  GEOMETRY 

Prof.  Halsted's  great  activity  has  never  less- 
ened. The  numerous  works  that  bear  his  name 
are  evidence  of  this.  Many  of  these  have  for 
their  aim  the  popularization  of  science,  and  are 
scattered  through  the  Monist,  the  Educational 
Review,  Popular  Science  Monthly,  etc. 

Among  his  text-books  (2)  besides  that  already 
mentioned,  his  Elementary  Synthetic  Geometry 
[New  York.  J.  Wiley  &  Sons],  deserves  special 
attention.  In  this  the  principal  points  of  Syn- 
thetic Geometry  are  brought  together  logically, 
and  rigorously  demonstrated.  This  is  one  of  the 
most  notable  books  in  American  didactic  literature, 
as  the  following  from  an  impartial  and  authorita- 
tive review  [Bulletin  of  the  New  York  Mathe- 
matical Society]  testifies  :  "For  more  than  2000 
years  geometry  has  been  founded  upon,  and  built 
up  by  means  of,  congruent  triangles.  At  last 
comes  a  book  reaching  all  the  preceding  results 
without  making  any  use  of  congruent  triangles; 
and    so    simply    that,   for    example,    all    ordinary 


(21  Others  of  these  are  for  example:  Elements  of  Ge- 
ometrv,  (J.  Wiley  &  Sons,  New  York).  Projective  Ge- 
ometry, (Ibid.) 


HALSTED'S  RATIONAL  GEOMETRY  67 

cases  of  congruence  of  triangles  are  demonstrated 
together  in  eigiit  lines." 

The  name  of  Halsted  is  indissolubly  connected 
with  the  popularization  of  non-Euclidean  geome- 
try (4).     From  the  time  when    he  fust    devoted 

(4)     The  following  publications  of  his  bear   upon   the 

popularization  of  Metageometry  : 

Gauss  and  the  Non-Euclidean  Geometry,— Science,  N.S., 
t.  XIV,  pag.  705-717  (1890);  The  Appreciation  of 
Non-Euclidean  Geometry, -ibid.,  pag.  462-465. 

Lambert's  Non-Euclidean  Geometry,— Bui.  of  the  New- 
York  Math,  Society,  t.  Ill,  pag.  78-80,  ( 1894). 

Non-Euclidean  Geometry ;  Historical  and  Expository,— 
American  Math.  Monthly,  t.  1,  H,  111,  (1894,  95,  96). 

The  Non-Eu:lidean  Geometry  inevitable,— The  Monist, 
t.  IV,  Chicago,  1894. 

Some  salient  points  in  the  History  of  Non-Euclidean  Ge- 
ometry and  Hyper-Spaces,— Math.  Papers  read  at  the 
Internat.  Math.  Congress,— Chicago,  1893. 

Nicolai  1.  Lobatchefsky— Address  pronounced  at  the  com- 
memorative meeting  of  the  Imperial  University  of 
Kasan,  October  22,  1893,  by  A.  Vassiliev (translated 
from  the  Russian,  with  a  preface).    1894. 

Darwinism  and  Non-Euclidean  Geometry,— Boll,  di  Ka- 
san, (2),  t.  VI,  pag.  25-29,  (1896). 

The  Introduction  to  Lobatchefsky's  new  elements  of 
Geometry,— Texas-Academy,  1897, 

Scientific  Books,  Urkunden   zur  Geschichte,— Science,  N. 

S.  t.  IX,  pag.  813-817,  (1889). 
Report  on  Progress   in    Non-Euclidean  Geometry,— Proc. 
of  the  Amer.  Ass.  f.  adv.  of  Sc.  t.  XLVllI,  pag.53-68 
(1899). 


68  HALSTED'S  RATIONAL  GEOMETRY 


himself  to  philosophical  studies  he  has  been  inter- 
terested  in  this  geometry,  becoming  not  only  its 
student  but  also  its  most  enthusiastic  apostle,  and 
winning  the  distinction  of  giving  its  first  bibliog- 
raphy, (1)  which  was  of  so  much  importance  as 
to  be  at  once  translated  and  reprinted  in  Russia. 
A  few  years  before  had  begun  in  Europe  tliat 
scientific  period  in  which  the  attention  of  geome- 
ters was  directed  to  researches  relating  to  the 
foundations  of  geometry,  and  to  that  geometry 
for  which  Sylvester  proposed  the  name  of  iiltra- 
Eiididean.  J.  Hoiiel  in  France  and  the  immortal 
Beltrami  in  Italy  were  almost  alone  in  empha- 
sizing the  high  importance  of  the  labors  of  Lo- 
bachevski  and  of  Bolyai,  and  in  pointing  out  that 
the  study  of  the  foundations  of  science  ought  to 

Non-Euclidean  Geometry,— Am.  Math.  Month.,  t.  Vll, 
pag.  123-133,  (1900). 

Non-Euclidean  Geometry  for  Teachers,— Popular  As- 
tronomy, 1900. 

Supplementary  Report  on  Non-Euclidean  Geometry,— 
Science,  N.  S.  t.  XIV,  pag.  705-717,  U9oO- 

The  Teaching  of  Geometry,— Educational  Review,  New 
York,  Dec.  1902,  pag.456-470. 

(i)  Bibliography  of  Hyperspace  and  Non-Euclidean 
Geometry,— Amer.  Jour,  of  Math.  vol.  1,  pag.  261-266  and 
384-385,  (1878,;  vol.  11,  pag.  65-70,  (1879). 


r 


HALSTED'S  RATIONAL  GEOMETRY  69 

rise  to  the  highest  philosophical  importance,  being 
perhaps  the  only  thing  capable  of  furnishing  the 
key  to  the  origin  and  formation  of  the  human 
consciousness.  What  marvel,  then,  that  in  young 
America  no  one  should  yet  have  put  himself  at 
the  head  of  those  who  aspired  to  be  attached  to 
the  school  that  had  shown  that  the  geometry 
which  for  more  than  2000  years  had  been  regarded 
as  the  only  possible  one  could  not  resist  a  serious 
investigation  of  its  postulates,  and  that  other  sys- 
tems of  geometry  just  as  rigorous  were  possible? 

However,  more  than  one  had  there  begun  to 
discuss  the  two  propositions  of  Legendre,  the  dem- 
onstrations of  which  involve  the  postulate  of  Ar- 
chimedes, and  had  shown  what  this  geometry 
would  be  without  this  postulate,  and  the  geometry 
of  Euclid  without  the  XI  axiom.  All  this,  to  be 
sure,  remained  in  the  exclusive  possession  of  the 
learned,  or  rather,  of  the  few  who  had  been 
initiated  in  the  new  studies.  Professor  Halsted 
undertook  the  work  of  placing  the  new  studies 
within  reach  of  all,  translating  the  works  of  the  Rus- 
sian Lobachevski  (1),  the  Hungarian  Bolyai  (2), 
the  Italian  Saccheri(3).  It  was  his  enthusiasm  that 

(i)     N.  Lobachevski,— Geometrical  Researches  on  the 


70         HALSTED'S  RATIONAL  GEOMETRY 


drew  many  into  the  way  marked  out  by  these 
heights,  and  very  soon  a  distinguished  band  of 
eminent  names  enriched  the  company  of  the  culti- 
vators of  the  new  ideas.  Familiarized  with  the 
labors  of  the  non-Euclidean  geometers,  fully  im- 
bued with  philosophic  tradition,  dominated  by 
critical  spirit  of  rare  vigor,  they  also  contributed  to 
make  evident  the  errors  and  philosophical  contra- 
dictions of  paradoxers,  and  to  overcome  the  unjust 
and  very  often  ignorant  objections  of  the  philoso- 
phers to  metageometry.  They  also  ranged  them- 
selves  among  those  who  wished  to  restore  and 

Theory  of  Parallels— (translated  from  the   original   with 
preface  and   appendix),— Tokvo  Sugakubutsurigiku  Ka- 
wai  Kiji,  t.  V,  pag.  6-50.  (1894). 

Ibid. — 4th  ed.  Austin  1894. 

Ibid.— Introduction   to    New   Elements  of  Geometry, 

with   a  complete    Theory  of    Parallels,— (translated 

from  the  Russian).     Austin,  1897. 

(2)  J.  Bolyai,— The  Science  absolute  of  Space,  inde- 
pendent of  the  truth  or  falsity  of  Euclid's  Axiom, — 
(translated  from  the  Latin).— Austin,  1894,  and  repro- 
duced also  in  Tokyo  Sugaku ,  t,  V.  pag.  94-134,(1894). 

(3)  nuclides  ab  omnI  naevo  vindicatus,  sive  conatus 
geometricus  quo  stabiliuntur  prima  ipsa  universae  Geo- 
metriae  principia, — Auctore  Hieronymo  Saccherio,  Socie- 
tate  Jesu,  in  Ticinensi  Universitate  Matheseos  Professore 
— Mediolani.     1733. 


HALSTED'S  RATIONAL  GEOMETRY  71 

correct  the  critique  theories,  showing  themselves 
disciples  and  continuers  of  Kant,  making  synthesis 
of  every  anterior  research  into  the  theory  of  groups 
which  enabled  Sophus  Lie  to  reduce  the  axioms 
of  geometry  to  their  logical  essence. 

How  deserved  is  the  gratitude  which  science 
owes  to  Halsted  is  shown  by  an  appreciation  of 
him  by  the  illustrious  Prof.  A.  Vassilief  of  the 
University  of  Kasan  in  a  letter  received  from  him 
a  few  days  ago: 

"  in  the  history  of  the  diffusion  of  the  ideas  of 
non-Euclidean  geometry,  the  name  of  Halsted  will 
always  be  mentioned  with  great  respect.  He  gave 
the  first  bibliography  of  the  works  on  non-Euclid- 
ean geometry.  He  gave  his  eminent  support 
to  the  work  of  the  Lobachevski  committee  estab- 
lished at  Kasan  in  1893  for  the  purpose  of  honor- 
ing the  memory  of  the  great  Russian  geometer. 
He  has  translated  into  English  various  works  of 
Lobachevski,  and  has  also  in  a  series  of  articles 
always  interesting  made  the  Anglo-Saxon  scientific 
world  acquainted  with  the  latest  literature  of  non- 
Euclidean  geometry.  The  indefatigable  zeal  with 
which  the  distinguished  professor  has  occupied 
himself  with  all  that  is  related  to  non-Euclidean 


72         HALSTED'S  RATIONAL  GEOMETRY 

geometry  is  derived  from  tine  philosophical  and 
gnoseological  interest  it  has  for  him.  He  has 
most  lucidly  set  forth  this  interest  in  his  article, 
'Darwinism  and  Non-Euclidean  Geometry,'  writ- 
ten at  my  request  during  his  stay  at  Kasan,  of 
which  I  shall  always  retain  the  pleasantest  mem- 
ory. The  long  journey  from  Texas  to  the  Volga, 
made  for  the  sole  purpose  of  honoring  the  memory 
ofJ^ob^ir^^^'^kL  's  also  proof  of  the  love — per- 
haps one  might  say  enthusiastic  devotion — Prof. 
Halsted  has  for  this  branch  of  geometrical  science. 
But  without  such  devotion  nothing  great  can  be 
done.  It  is  assured  that  American  scientific  liter- 
ature will  soon  receive  from  Prof.  Halsted  a  com- 
plete history  of  Non-Euclidean  geometry,  which 
we  do  not  yet  possess.  it  will  be  a  fitting  culmi- 
nation of  his  labors  to  propagate  in  Anglo-Ameri- 
can literature  the  ideas  of  Lobachevski  and  of 
Bolyai." 

And  it  is  precisely  this  which  1  also  presage  in 
presenting  to  the  noble  professor  the  warmest 
well-wishing  and  the  most  affectionate   salutation. 

Prof.  C.  Alasia. 
tempio  (sardinia), 

March,  1903. 


HALSTED'S  RATIONAL  GEOMETRY  73 


COMMENTS  OF  MATHEMATICIANS 

Sehr  geehrter  Herr  College! 

Der  Internationale  Mathematikercongress  zu 
Heidelberg,  dem  ich  in  voriger  Woche  beiwohnte, 
hat  mich  bisher  verhindert,  Ihren  fiir  Ihr  schones 
Text-book  on  Rational  Geometry  den  Dank  auszu- 
sprechen,  der  auch  ohne  ihr  liebenswiirdiges 
Schreiben  vom  1.  8.  sehr  bald  erfolgt  ware. 

Ic'ii  habe  ihr  Buch  mit  dem  grossten  Interesse 
gelesen  und  mich  gefreut,  dass  wir  nun  endlich 
eine  Elementargeometrie  besitzen  in  der  die  Pro- 
portionslehre  ohne  das  Archimedische  Postulat 
entwickelt  ist. 

Ich  bin  sicher,  das  ihr  Buch  ein  Vorbild  sein 
wird  fiir  viele  andre,  die  je  nach  den  ortlichen  Be- 
diirfnissen  in  andern  Landern  verfasst  werden 
werden. 

Darf  ich  mir  eine  Bemerkung  erlauben,  so  ist 
es  die,  dass  nicht    recht    ersichtlich   ist,  wozu  Sie 


74  HALSTED'S  RATIONAL  GEOMETRY 


das  Archimedische  Postulat  iiberhaupt  anfiihren, 
da  Sie  es  weder  in  dtzT  Lehre  vom  In  halt  der  Poly- 
gone  und  Polyeder  noch  zum  Beweise  der  As- 
sumption VI  1,  p.  259,  beniitzen.  Aber  vielleicht 
sind  mir  die  Stellen,  an  denen  es  gebraucht  wird, 
entgangen. 

Ihre  Lehre  vom  Volumen  der  Polyeder  hat  mich 
um  so  mehr  interessiert,  als  ich  selber  friiher 
einen  ahnlichen  Versuch  gemacht  habe. 

Ich  war  aber  doch  nicht  so  ganz  davon  befrie- 
digt,  dass  der  Begriff  des  Volumens  ganzlich  von 
dem  der  ^quivalenz  abgelGst  wurde,  mochte  das 
auch  ohne  das  Archimedische  Postulat  nicht 
moeglich  sein.  Ich  erkenne  aber  die  Berechtigung 
Ihrer  Auffassung  vollkommen  an. 

Dass  die  Assumption  VI  1  sich  aus  dem  Ar- 
chimedische Postulat  beweisen  lasst,  ist  Ihnen 
gewiss  nicht  entgangen,  vielleicht  aber  bemerken 
Sie  nicht,  dass  die  Assumption  VI  2  sich  ganz 
einfach  ohne  Beni^itzung  des  Archimedische  Pos- 
tulat auf  VI  1  zuriickfiihren  lasst. 

Indem  ich  Ihnen  nochmals  meinen  besten  Dank 
fiir  den  Genuss  ausspreche,  den  Sie  mir  durch 
die  Ueberreichung    Ihres    Buch   verschafft  haben, 


HALSTED'S  RATIONAL  GEOMETRY  75 


zelchne    ich    mit    dem  Ausdrucke  def  Hochschat- 
zung  als  Ihr  ergebenster 

F.   SCHUR. 
FREUDENSTADT, 
1 8.  8.  04. 

[From  Professor  Friedriech  Schur  of  Karlsruhe,  one  of 
two  greatest  living  authorities  on  elementary  geometry.] 

My  Dear  Dr.  Halsted: 

1  have  just  received  your  letter  and  a  day  or  so 
since  the  copy  of  the  Rational  Geometry.  It  is 
an  excellent  piece  of  work  and  will  do  much  good, 
I  am  sure. 

It  is  certainly  a  very  necessary  thing  to  have 
the  scientific  truths  of  geometry  put  into  such 
perfect  shape  and  so  available  for  the  understand- 
ing of  students.  This,  it  seems  to  me,  is  the  true 
popularizing  of  mathematics. 

Your  old  teacher  Sylvester  would  rejoice,  I 
know,  in  the  work  you  have  been  doing. 

I  thank  you  ever  so  much  for  the  copy  of  your 
book  and  also  in  behalf  of  mathematical  teaching 
in  this  country.         Yours  most  truly, 

W.  H.  Echols. 

Professor  of  Mathematics  in  the 
University  of  Virginia. 
UNIVERSITY  OF  VIRGINIA, 
August  5,  1904. 


76       halsted's  rational  geometry 

My  Dear  Professor  Halsted: 

Your  Rational  Geometry  is  a  beautiful  piece  of 
work  which  in  my  opinion  is  destined  to  have  a 
marked  influence  on  the  teaching  of  elementary 
geometry. 

1  think  every  teacher  of  geometry  should  make 
a  careful  study  of  this  book. 

Yours  very  truly, 

P.  A.  Lambert. 

Professor  of  Mathematics  in  Lehigh  University. 

BETHLEHEM,  PA., 
Aug.  i6,   1904. 

Messrs.  John  Wiley  &  Sons: 

Dear  Sirs — Halsted 's  Rational  Geometry  con- 
stitutes a  new  departure,  and  its  production  is 
eminently  characteristic  of  its  author.  His  aim 
marks  an  epoch  in  the  teaching  of  the  subject  in 
this  country.  All  teachers  would  be  greatly  prof- 
ited by  its  perusal.  If  the  influx  of  new  ideas  in 
geometry  is  to  produce  an  early  effect  in  the  coun- 
try as  a  whole,  it  will  have  to  do  it  through 
reaching  the  teachers.  For  such  a  use  the  Ra- 
tional Geometry  is  emniently  well  adapted. 

If  such  works   as  the    Rational  Geometry  and 


HALSTED'S  RATIONAL  GEOMETRY  77 


Professor  Halsted's  several  productions  on  the 
non-Euclidean  geometry  receive  reading  by  our 
teachers  of  geometry,  the  educational  effect  will 
most  likely  be  far  greater  than  if  these  works  had 
a  limited  actual  use  in  our  schools.  Here's  hoping 
that  the  youth  of  our  country  will  get  the  new 
ideas  and  ideals  through  the  medium  of  their 
teachers.  Yours  very  truly, 

JOS.  V.  Collins. 

State  normal  school, 
Stevens  Point,  wis., 

Oct.  21,   1994. 

My  Dear  Dr.  Halsted: 

1  have  looked  over  your  Rational  Geometry 
with  great  interest.  The  book  should  be  read  by 
every  teacher.  Very  sincerely, 

H.   E.   HAWKES. 

Professor  of  Mathematics,  Yale  University. 
20CARMEL  ST.,  NEW  HAVEN. 

Professor  Halsted's  Rational  Geometry  is  a 
book  that  every  teacher  and  student  of  mathe- 
matics should  possess.  It  combines  clearness  and 
simplicity  with  rigor,  in  which  last  quality  Euclid, 
and  still  more  some  of  his  modern  rivals,  are  sadly 
deficient.     It  takes  into   account  and    utilizes  the 


78  HALSTED'S  RATIONAL  GEOIVIETRY 

results  of  all  the  centuries  of  inv^estigation  since 
Euclid;  in  fact  a  book  like  this  is  unthinkable 
without  Lobachevsky,  Bolyai  or  Hilbert.  No 
tacit  assumptions  as  in  Euclid,  nostraight-line-the- 
shortest-distance  "axiom"  as  in  most  of  our  mod- 
ern text-books,  no  doubtful,  erroneous  and  irrele- 
vant statements  concerning  non-Euclidean  geome- 
tries as  in  some  other  text-books,  but  a  reliable, 
complete  and  rigorous  system  of  geometry  such 
as  could  have  been  written  only  after  the  modern 
investigations  on  the  foundation  of  geometry  had 
been  concluded  in  their  essential  features.  It  is 
the  first  book  of  its  kind  in  our  country  and  in 
any  country  (Italy  perhaps  excepted),  and  this 
fact  alone  makes  good  its  claim  to  the  attention  of 
teachers  and  students  of  mathematics. 

JOHN    ElESLAND. 
Instructor,  U.  S.  Naval  Academy,  Annapolis,  Md. 

My  Dear  Professor  Halsted: 

For  simplicity  of  form  and    rigor  of  logic    your 
book  is  a  veritable  model. 

Yours  very  truly,         ARNOLD  Emch. 

Professor  of  Mathematics,  Univ.  of  Colorado. 
Nov.  21,  1904. 


halsted's  rational  geometry        79 

My  Dear  Professor  Halsted: 

Your  Rational  Geometry  came  from  the  pub- 
lisliers  some  days  ago.  1  have  read  consecutively 
the  first  ten  chapters. 

Its  simple,  rigorous  logic,  its  accurate,  concise, 
terse  English  mark  your  book  as  a  masterpiece  of 
geometrical  exposition. 

It  appeals  to  me  as  simpler  and  easier  than  the 
usual  text.  Boys  and  girls  who  are  ready  for 
demonstrative  geometry  should  have  no  difficulty 
with  it. 

You  have  done  them  and  the  teachers  and  the 
cause  of  mathematics  in  this  country  a  great 
service. 

1  am  delighted  with  the  book.  If  my  studies 
entitled  my  opinion  to  any  weight  in  such  matters 
(and  they  do  not)  I  should  say  that  your  book  is 
the  most  important  contribution  to  the  text-book 
literature  of  elementary  geometry  since  Euclid. 
Most  truly  yours, 

Thos.  E.  McKinney. 

Professor  of  Mathematics  in  Marietta  College. 
Secretary  of  the  Association  of  Ohio  Teachers 
of  Mathematics  and  Science. 

MARIETTA,  OHIO, 
Aug.  4,   1904- 


80  HALSTED'S  RATIONAL  GEOMETRY 

From  the  Preface  written  by  Poincare  for  the 
American  edition  of  his  "Science  and  Hypoth- 
esis." 

Je  SLiis  tres  reconnaissant  a  IW.  Halsted  qui  a 
bien  voulu,  dans  une  traduction  claire  et  fidele, 
presenter  mon  livre  aux  lecteurs  americains.  On 
sait  que  ce  savant  a  deja  pris  la  peine  de  traduire 
beaucoup  d'ouvrages  europeens  et  a  ainsi  puis- 
samment  contribue  a  faire  m.ieux  connaitre  au  no- 
veau  continent  la  pensee  de  I'ancien.   .   .   . 

D'ailleurs  M.  Halsted  donne  regulierement 
chaque  annee  une  revue  des  travaux  relatifs  a  la 
geometrie  non-euclidienne,  et  il  a  autour  de  lui  un 
public  qui  s'interesse  a  son  oeuvre. 

II  a  initie  ce  public  aux  idees  de  M.  Hiibert  et  il 
a  meme  ecrit  un  traite  elementaire  de  Rational 
Geometry,  fonde  sur  les  principes  du  celebre  sa- 
vant allemand. 

Introduire  ce  principe  dans  I'enseignement,  c'est 
bien  pour  le  coup  rompre  les  ponts  avec  I'intuition 
sensible,  et  c'est  la  je  I'avoue,  une  hardiesse  qui 
me  parait  presque  une  temerite. 

Le  public  americain  est  done  beaucoup  mieux 
prepare  qu'on  ne  le  pense  a  rechercher  I'origine 
de  la  notion  d'espace. 


14  DAY  USE 


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